---
abstract: 'We investigate the asymptotic behavior of solutions of anisotropic equations of the form $-\sum_{i=1}^n\partial_{x_i}\big(\left|\partial_{x_i}u\right|^{p_i-2}\partial_{x_i}u\big)=f{\left(}x,u{\right)}$ in ${\mathbb{R}}^n$, where $p_i>1$ for all $i=1,\dotsc,n$ and $f$ is a Caratheodory function with critical Sobolev growth. This problem arises in particular from the study of extremal functions for a class of anisotropic Sobolev inequalities. We establish decay estimates for the solutions and their derivatives, and we bring to light a vanishing phenomenon which occurs when the maximum value of the $p_i$ exceeds a critical value.'
address: 'Jérôme Vétois, Univ. Nice Sophia Antipolis, CNRS, LJAD, UMR 7351, 06108 Nice, France.'
author:
- Jérôme Vétois
date: 'October 2, 2014. [*Final version:*]{} July 22, 2015.'
title: Decay estimates and a vanishing phenomenon for the solutions of critical anisotropic equations
---

[^1]

Introduction and main results {#Sec1}
=============================

We let $n\ge2$ and $\operatorname{\overrightarrow{p}}={\left(}p_1,\dotsc,p_n{\right)}$ be such that $p_i>1$ for all $i=1,\dotsc,n$ and $\sum_{i=1}^n1/p_i>1$. In this paper, we are interested in the solutions of problems of the form $$\label{Eq1}
\left\{\begin{aligned}&-\Delta_{\operatorname{\overrightarrow{p}}} u=f{\left(}x,u{\right)}\quad\text{in }{\mathbb{R}}^n,\\
&u\in D^{1,\operatorname{\overrightarrow{p}}}{\left(}{\mathbb{R}}^n{\right)},
\end{aligned}\right.$$ where $\Delta_{\operatorname{\overrightarrow{p}}} u:=\sum_{i=1}^n\partial_{x_i}{\left(}\left|\partial_{x_i}u\right|^{p_i-2}\partial_{x_i}u{\right)}$ is the anisotropic Laplace operator, $D^{1,\operatorname{\overrightarrow{p}}}{\left(}{\mathbb{R}}^n{\right)}$ is the completion of $C^\infty_c{\left(}{\mathbb{R}}^n{\right)}$ with respect to the norm $\left\|u\right\|_{D^{1,\operatorname{\overrightarrow{p}}}{\left(}{\mathbb{R}}^n{\right)}}:=\sum_{i=1}^n{\left(}\int_{{\mathbb{R}}^n}\left|\partial_{x_i}u\right|^{p_i}dx{\right)}^{1/p_i}$, and $f:{\mathbb{R}}^n\times{\mathbb{R}}\to{\mathbb{R}}$ is a Caratheodory function such that $$\label{Eq2}
\left|f{\left(}x,s{\right)}\right|\le\Lambda\left|s\right|^{p^*-1}\quad\text{for all }s\in{\mathbb{R}}\text{ and a.e. }x\in{\mathbb{R}}^n,$$ for some real number $\Lambda>0$. Here, $p^*$ denotes the critical Sobolev exponent and is defined as $$p^*:=\frac{n}{\sum_{i=1}^n\frac{1}{p_i}-1}=\frac{np}{n-p}\qquad\text{with}\quad\frac{1}{p}:=\frac{1}{n}\sum_{i=1}^n\frac{1}{p_i}\,.$$

The problem with $f{\left(}x,u{\right)}=\left|u\right|^{p^*-2}u$ appears in the study of extremal functions for a class of anisotropic Sobolev inequalities. Early references on anisotropic Sobolev inequalities are Nikolski[ĭ]{} [@Nik], Troisi [@Tro], and Trudinger [@Tru]. We also refer to Cianchi [@Cia1] for a more recent work on the topic. Here we are interested in an inequality which appeared first in Troisi [@Tro]. Among different equivalent versions (see Theorem \[Th4\] below), this inequality can be stated as $$\label{Eq3}
\int_{{\mathbb{R}}^n}\left|u\right|^{p^*}dx\le C{\left(}\sum_{i=1}^n\int_{{\mathbb{R}}^n}\left|\partial_{x_i}u\right|^{p_i}dx{\right)}^{p^*/p}$$ for some constant $C=C{\left(}n,\operatorname{\overrightarrow{p}}{\right)}$ and for all functions $u\in C^\infty_c{\left(}{\mathbb{R}}^n{\right)}$. The inequality enjoys an anisotropic scaling law (see below). As a corollary of the work of El Hamidi–Rakotoson [@ElHRak], we obtain in Theorem \[Th5\] below that there exist extremal functions for the inequality provided that $p_i<p^*$ for all $i=1,\dotsc,n$.

In the presence of anisotropy, namely when the $p_i$ are not all equal, there is no explicit formula for the extremal functions of . This motivates to find a priori estimates for these functions, and more generally for the solutions of equations of type . The main difficulties in this work come from the non-homogeneity of the problem and the lack of radial symmetry.

As a more general motivation, the solutions of problems of type with $f{\left(}x,u{\right)}=\left|u\right|^{p^*-2}u$ turn out to play a central role in the blow-up theories of critical equations in general domains. Possible references in book form on this subject and its applications in the isotropic regime are Druet–Hebey–Robert [@DruHebRob], Ghoussoub [@Gho], and Struwe [@Str2]. A first step in the direction of a blow-up theory in the anisotropic regime was taken in El Hamidi–Vétois [@ElHVet] where we extended the bubble tree decompositions of Struwe [@Str1]. Now, if one wants to go further and investigate a pointwise blow-up theory, then it is essential to know the asymptotic behavior of the solutions of with $f{\left(}x,u{\right)}=\left|u\right|^{p^*-2}u$. The results in this paper can be seen as a crucial step in this direction.

Anisotropic equations of type have received much attention in recent years. In addition to the above cited references [@ElHRak; @ElHVet] and without pretending to be exhaustive, we mention for instance the works by Cianchi [@Cia2] on symmetrization properties, Cîrstea–Vétois [@CirVet] on the fundamental solutions, Cupini–Marcellini–Mascolo [@CupMarMas] on the local boundedness of solutions, Fragalà–Gazzola–Kawohl [@FraGazKaw] on the existence and non-existence of solutions in bounded domains, Lieberman [@Lie] on gradient estimates, Namlyeyeva–Shishkov–Skrypnik [@NamShiSkr] on singular solutions, and Vétois [@Vet2] on vanishing properties of solutions. More references can be found for instance in [@Vet2].

Throughout this paper, we denote $$\label{Eq4}
p_+:=\max{\left(}\left\{p_i\in\operatorname{\overrightarrow{p}}\right\}{\right)}\quad\text{and}\quad p_*:=\frac{n-1}{\sum_{i=1}^n\frac{1}{p_i}-1}=\frac{p{\left(}n-1{\right)}}{n-p}\,.$$ The exponent $p_*$ is known to play a critical role in several results on the asymptotic behavior of solutions of second order elliptic equations (see the historic paper of Serrin [@Ser], see also for instance the more recent paper of Serrin–Zou [@SerZou] and the references therein).

Our first result is as follows.

\[Th1\] Assume that $p_+<p_*$. Let $f:{\mathbb{R}}^n\times{\mathbb{R}}\to{\mathbb{R}}$ be a Caratheodory function such that holds true and $u$ be a solution of . Then there exists a constant $C_0=C_0{\left(}n,\operatorname{\overrightarrow{p}},\Lambda,u{\right)}$ such that $$\label{Th1Eq1}
\left|u{\left(}x{\right)}\right|^{p_*}+\sum_{i=1}^n\left|\partial_{x_i}u{\left(}x{\right)}\right|^{p_i}\le C_0\bigg(1+\sum_{i=1}^n\left|x_i\right|^{\frac{p_*p_i}{p_*-p_i}}\bigg)^{-1}\quad\text{for a.e. }x\in{\mathbb{R}}^n,$$ where $p_*$ is as in .

We point out that the decay rate in is the same as the one obtained in Cîrstea–Vétois [@CirVet] for the fundamental solutions in ${\mathbb{R}}^n$, namely the solutions of the equation $-\Delta_{\operatorname{\overrightarrow{p}}}u=\delta_0$ in ${\mathbb{R}}^n$, where $\delta_0$ is the Dirac mass at the point 0.

In case all $p_i$ are equal to $p$, as part of a more general result, Alvino–Ferone–Trombetti–Lions [@AlvFerTroLio] proved that the best constant in the inequality is attained by the functions $$\label{Eq5}
u_{a,b}{\left(}x{\right)}:=\bigg(a+b\sum_{i=1}^n\left|x_i\right|^{\frac{p}{p-1}}\bigg)^{\frac{p-n}{p}}$$ for all $a,b>0$. Moreover, Cordero-Erausquin–Nazaret–Villani [@CorNazVil] proved that the functions are the only extremal functions of . In case where the norm of the gradient in is replaced bv the Euclidean norm, the existence of radially symmetric extremal functions was found by Aubin [@Aub], Rodemich [@Rod], and Talenti [@Tal]. Since $p/{\left(}p_*-p{\right)}={\left(}n-p{\right)}/{\left(}p-1{\right)}$, the decay rate in coincides with the one in .

In case of the Laplace operator ($p_i=2$), Caffarelli–Gidas–Spruck [@CafGidSpr] (see also Chen–Li [@ChenLi]) proved that every positive solution of with $f{\left(}x,u{\right)}=u^{p^*-1}$ is of the form . This result can be extended to the case where all $p_i$ are equal to $p\in{\left(}1,n{\right)}$ for positive solutions satisfying the one-dimensional symmetry $u{\left(}x{\right)}=u\big(\sum_{i=1}^n\left|x_i\right|^{\frac{p}{p-1}}\big)$ for all $x\in{\mathbb{R}}^n$. Indeed, this result has been proved by Guedda–Véron [@GueVer] in case of positive, radially symmetric solutions for the $p$–Laplace equation $-\operatorname{div}{\left(}\left|\nabla u\right|^{p-2}\nabla u{\right)}=u^{p^*-1}\quad\text{in }{\mathbb{R}}^n$, and it can easily be seen that both cases lead to the same ordinary differential equation. We also mention that radial symmetry results have been established for positive solutions in $D^{1,p}{\left(}{\mathbb{R}}^n{\right)}$ in the case of $p$–Laplace equations (see Damascelli–Merchán–Montoro–Sciunzi [@DamMerMonSci], Damascelli–Ramaswamy [@DamRam], Sciunzi [@Sci], and Vétois [@Vet3]).

Theorem \[Th1\] has been proved in Vétois [@Vet3] in case of the $p$–Laplace operator. We also refer in case of the Laplace operator ($p_i=2$) to Jannelli–Solimini [@JanSol], where the decay estimate has been proved to hold true for solutions of with right-hand side $f{\left(}x,u{\right)}=\sum_{i=1}^Na_i{\left(}x{\right)}\left|u\right|^{q_i^*-2}u$, where $q_i^*:=2^*{\left(}1-1/q_i{\right)}$, $q_i\in{\left(}n/2,\infty{\right]}$, $\left|a_i{\left(}x{\right)}\right|={\text{O}}\big(\left|x\right|^{-n/q_i}\big)$ for large $\left|x\right|$, and $a_i$ belongs to the Marcinkiewicz space $M^{q_i}{\left(}{\mathbb{R}}^n{\right)}$ for all $i=1,\dotsc,N$.

The next results concern the case $p_+\ge p_*$, namely $p_i\ge p_*$ for some index $i$. In particular, we are now exclusively in the case where the exponents $p_i$ are not all equal.

In the limit case $p_+=p_*$, we prove the following result.

\[Th2\] Assume that $p_+=p_*$. Let $f:{\mathbb{R}}^n\times{\mathbb{R}}\to{\mathbb{R}}$ be a Caratheodory function such that holds true and $u$ be a solution of . Then for any $q>p_*$, there exists a constant $C_q=C{\left(}n,\operatorname{\overrightarrow{p}},\Lambda,u,q{\right)}$ such that $$\label{Th2Eq}
\left|u{\left(}x{\right)}\right|^{q}+\sum_{i=1}^n\left|\partial_{x_i}u{\left(}x{\right)}\right|^{p_i}\le C_q\bigg(1+\sum_{i=1}^n\left|x_i\right|^{\frac{qp_i}{q-p_i}}\bigg)^{-1}\quad\text{for a.e. }x\in{\mathbb{R}}^n.$$

Beyond this limit case, namely when $p_*<p_+<p^*$, we find the following result.

\[Th3\] Assume that $p_*<p_+<p^*$. Let $f:{\mathbb{R}}^n\times{\mathbb{R}}\to{\mathbb{R}}$ be a Caratheodory function such that holds true and $u$ be a solution of . Then there exist a real number $q_0=q_0{\left(}n,\operatorname{\overrightarrow{p}}{\right)}<p_+$ such that the two following assertions hold true.

1.  There exists a constant $R_0=R_0{\left(}n,\operatorname{\overrightarrow{p}},\Lambda,u{\right)}$ such that $$\label{Th3Eq1}
    u{\left(}x{\right)}=0\quad\text{for all }x\in{\mathbb{R}}^n\text{ such that }\sum_{i\in\mathcal{I}_0}\left|x_i\right|\ge R_0\,,$$ where $\mathcal{I}_0$ is the set of all indices $i$ such that $p_i>q_0$. Moreover, $\mathcal{I}_0\ne\emptyset$ due to $q_0<p_+$.

2.  For any $q>q_0$, there exists a constant $C_q=C{\left(}n,\operatorname{\overrightarrow{p}},\Lambda,u,q{\right)}$ such that $$\label{Th3Eq2}
    \left|u{\left(}x{\right)}\right|^{q}+\sum_{i=1}^n\left|\partial_{x_i}u{\left(}x{\right)}\right|^{p_i}\le C_q\bigg(1+\sum_{i\in\mathcal{I}_0^c}\left|x_i\right|^{\frac{qp_i}{q-p_i}}\bigg)^{-1}\quad\text{for a.e. }x\in{\mathbb{R}}^n,$$ where $\mathcal{I}_0^c:=\left\{1,\dotsc,n\right\}\backslash\mathcal{I}_0$.

We are able, moreover, to give an explicit definition in terms of $n$ and $\operatorname{\overrightarrow{p}}$ of a real number $q_0$ satisfying the above result (see Section \[Sec7\]).

The dependence on $u$ of the constants $C_0$, $C_q$, and $R_0$ in the above results will be made more precise in Remarks \[Rem2\] and \[Rem3\].

As a remark about the support of solutions, by a result in Vétois [@Vet2], we have that for any nonnegative solution $u$ of with $f{\left(}x,u{\right)}$ as in (see [@Vet2] for the general assumptions), if $u{\left(}x{\right)}=0$ for some $x\in{\mathbb{R}}^n$, then we have $u\equiv0$ on the affine subspace $\left\{y\in{\mathbb{R}}^n:\,y_i=x_i\quad\forall i=\left\{1,\dotsc,n\right\}\backslash\mathcal{I}_-\right\}$, where $\mathcal{I}_-$ is the set of all indices $i$ such that $p_i=\min{\left(}\left\{p_j\in\operatorname{\overrightarrow{p}}\right\}{\right)}$. In case all $p_i$ are equal to $p$, we obtain that either $u>0$ or $u\equiv0$, thus recovering the same result as Vazquez [@Vaz] found for the $p$–Laplace operator. In the presence of anisotropy, as shows for instance Theorem \[Th3\], this result does not hold true in general on the whole ${\mathbb{R}}^n$.

We also point out that in the limit case $p_+=p^*$, we are able to construct quasi-explicit examples of solutions of with $f{\left(}x,u{\right)}=\left|u\right|^{p^*-2}u$ for anisotropic configurations of type $\operatorname{\overrightarrow{p}}={\left(}p_-,\dotsc,p_-,p_+,\dotsc,p_+{\right)}$ by using the method of separation of variables (see Vétois [@Vet1]). These solutions turn out to vanish in the $i$-th directions corresponding to $p_i=p_+$, exactly like what we prove to be true in Theorem \[Th3\] in case $p_*<p_+<p^*$.

The paper is organized as follows. In Section \[Sec2\], we present different equivalent versions of the anisotropic Sobolev inequality, and we study the existence and scaling properties of extremal functions for these inequalities.

Section \[Sec3\] is concerned with preliminary properties satisfied by the solutions of , namely global boundedness results and a weak decay estimate.

In Sections \[Sec4\] and \[Sec5\], we perform a Moser-type iteration scheme inspired from the one developed in Cîrstea–Vétois [@CirVet] for the fundamental solutions. In order to treat a large part of the proofs in a unified way, we consider a general family of domains defined as $$\label{Eq6}
\Omega_{{\overrightarrow{q}}}{\left(}\mathcal{I}_1,R_1,\mathcal{I}_2,R_2,\lambda{\right)}:=\bigg\{x\in{\mathbb{R}}^n:\sum_{i\in\mathcal{I}_1}\left|x_i\right|^{q_i}\hspace{-2pt}<\hspace{-2pt}{\left(}1+\lambda{\right)}R_1\text{ and }\bigg|\sum_{i\in\mathcal{I}_2}\left|x_i\right|^{q_i}-R_2\bigg|\hspace{-2pt}<\hspace{-2pt}\lambda R_2\bigg\},$$ where $\lambda\in{\left(}0,1{\right)}$, $R_1,R_2>0$, $\mathcal{I}_1$ and $\mathcal{I}_2$ are two disjoint subsets of $\left\{1,\dotsc,n\right\}$, $\mathcal{I}_2\ne\emptyset$, and ${\overrightarrow{q}}={\left(}q_i{\right)}_{i\in\mathcal{I}_1\cup\mathcal{I}_2}$ is such that $q_i>1$ for all $i\in\mathcal{I}_1\cup\mathcal{I}_2$. On these domains, we prove that the solutions of satisfy reverse Hölder-type inequalities of the form $$\label{Eq7}
\left\|u\right\|_{L^\gamma{\left(}\Omega_{{\overrightarrow{q}}}{\left(}\mathcal{I}_1,R_1,\mathcal{I}_2,R_2,\lambda{\right)}{\right)}}^\gamma\le C\max_{i\in\mathcal{I}_1\cup\mathcal{I}_2}\Big({\left(}\lambda'-\lambda{\right)}^{-p_i}\\
R_{\delta_i}^{-\frac{p_i}{q_i}}\left\|u\right\|_{L^{\gamma_i}{\left(}\Omega_{{\overrightarrow{q}}}{\left(}\mathcal{I}_1,R_1,\mathcal{I}_2,R_2,\lambda'{\right)}{\right)}}^{\gamma_i}\Big)^\frac{n}{n-p}$$ for all $\gamma>p_*-1$ and $\lambda<\lambda'\in{\left(}0,1/2{\right]}$, where $\delta_i:=1$ if $i\in\mathcal{I}_1$, $\delta_i:=2$ if $i\in\mathcal{I}_2$, $\gamma_i:=\frac{n-p}{n}\gamma+p_i-p$ for all $i\in\mathcal{I}_1\cup\mathcal{I}_2$, and $C=C{\left(}n,\operatorname{\overrightarrow{p}},{\overrightarrow{q}},u,\gamma{\right)}$ (see Lemma \[Lem4\] for more details on the dependence of the constant with respect to $u$ and $\gamma$). Since the right-hand side of involves different exponents $\gamma_i$ in the anisotropic case, the number of exponents in the estimates may grow exponentially when iterating this inequality. We overcome this issue in Section \[Sec5\] by controlling the values of the exponents with respect to the number of iterations.

In Section \[Sec6\], we prove a vanishing result which will give Point (i) in Theorem \[Th3\]. We prove this result by applying our iteration scheme with $R_1=R_2^{1/\varepsilon}$ for small real numbers $\varepsilon>0$ and $\mathcal{I}_1$, $\mathcal{I}_2$ being the sets of all indices $i$ such that $p_i<p_0$, $p_i=p_0$, respectively, for some large enough real number $p_0\in\operatorname{\overrightarrow{p}}$ (see for the exact condition on $p_0$). Passing to the limit into our iteration scheme, we obtain a pointwise estimate of the form $$\label{Eq8}
\left\|u\right\|_{L^\infty(\Omega_{{\overrightarrow{q}}}(\mathcal{I}_1,R^{1/\varepsilon},\mathcal{I}_2,R,1/4))}\le\big(CR^{-\frac{1}{p_\varepsilon}}\big)^{\frac{1}{\varepsilon}}$$ for some constant $C=C{\left(}n,\operatorname{\overrightarrow{p}},u{\right)}$ (see Lemma \[Lem8\]). When $R$ is large enough, the right-hand side of converges to 0 as $\varepsilon\to0$, and we thus obtain our vanishing result.

In Section \[Sec7\], we prove Theorems \[Th1\], \[Th2\], and we complete the proof of Theorem \[Th3\] by proving the decay estimates . The proofs of these results rely again on our iteration scheme, this time applied with $\mathcal{I}_1=\emptyset$ and $\mathcal{I}_2$ being the set of all indices $i$ such that $p_i\le\overline{p}_0$ for some real number $\overline{p}_0$ (see ).

Finally, in Appendix \[App\], we prove a weak version of Kato’s inequality which is used in Sections \[Sec3\] and \[Sec4\].

[**Acknowledgments.**]{} The author wishes to express his gratitude to Frédéric Robert for helpful comments on the manuscript.

Application to the extremal functions of a class of\
anisotropic Sobolev inequalities {#Sec2}
====================================================

As mentioned in the introduction, one of our main motivation in this paper is to apply our results to the extremal functions of a class of anisotropic Sobolev inequalities which originates from Troisi [@Tro]. In this section, we first present in Theorem \[Th4\] below different equivalent versions of these inequalities, and we then prove in Theorem \[Th5\] that all these inequalities have extremal functions, and that with a suitable change of scale, these extremal functions are solutions of with $f{\left(}x,u{\right)}=\left|u\right|^{p^*-2}u$.

We state the equivalent versions of the anisotropic Sobolev inequalities as follows.

\[Th4\] The following inequalities hold true.

1.  There exists a constant $C=C\big(n,\operatorname{\overrightarrow{p}}\big)$ such that $$\label{Th4Eq1}
    {\left(}\int_{{\mathbb{R}}^n}\left|u\right|^{p^*}dx{\right)}^{n/p^*}\le C\prod_{i=1}^n{\left(}\int_{{\mathbb{R}}^n}\left|\partial_{x_i}u\right|^{p_i}dx{\right)}^{1/p_i}\quad\forall u\in C^\infty_c{\left(}{\mathbb{R}}^n{\right)}.$$

2.  For any $\overrightarrow{\theta}={\left(}\theta_1,\dotsc,\theta_n{\right)}$ such that $\theta_i>0$ for all $i=1,\dotsc,n$ and $\sum_{i=1}^n1/\theta_i=n/p$, there exists a constant $C_{\overrightarrow{\theta}}=C\big(n,\operatorname{\overrightarrow{p}},\overrightarrow{\theta}\big)$ such that $$\label{Th4Eq2}
    {\left(}\int_{{\mathbb{R}}^n}\left|u\right|^{p^*}dx{\right)}^{p/p^*}\le C_{\overrightarrow{\theta}}\sum_{i=1}^n{\left(}\int_{{\mathbb{R}}^n}\left|\partial_{x_i}u\right|^{p_i}dx{\right)}^{\theta_i/p_i}\quad\forall u\in C^\infty_c{\left(}{\mathbb{R}}^n{\right)}.$$ In particular, we get in case $\theta_i=p_i$ for all $i=1,\dotsc,n$.

As a remark, the inequalities and enjoy an anisotropic scaling law. Indeed, it can easily be seen that every integral in these inequalities are invariant with respect to the change of scale $u\mapsto u_\lambda$, where $$\label{Eq9}
u_\lambda{\left(}x{\right)}=\lambda u{\left(}\lambda^{{\left(}p^*-p_1{\right)}/p_1}x_1,\dotsc,\lambda^ {{\left(}p^*-p_n{\right)}/p_n}x_n{\right)}$$ for all $\lambda>0$ and $x\in{\mathbb{R}}^n$.

We refer to Troisi [@Tro]\*[Theorem 1.2]{} for the proof of the inequality . Then the inequality follows from by applying an inequality of weighted arithmetic and geometric means. As a remark, we can also obtain from by applying the change of scale below.

Regarding the extremal functions of and , we prove the following result. The existence part in this result will be obtained as a corollary of the work of El Hamidi–Rakotoson [@ElHRak] and Proposition \[Pr1\] below.

\[Th5\] If $p_+<p^*$, then there exist extremal functions $u\in D^{1,\operatorname{\overrightarrow{p}}}{\left(}{\mathbb{R}}^n{\right)}$, $u\ne0$, of and . Moreover, for any extremal function $u$ of or , there exist $\mu_1,\dotsc,\mu_n>0$ such that the function $x\in{\mathbb{R}}^n\mapsto u{\left(}\mu_1x_1,\dotsc,\mu_nx_n{\right)}$ is a constant-sign solution of with $f{\left(}x,u{\right)}=\left|u\right|^{p^*-2}u$. In particular, every extremal function of or satisfies the a priori estimates in Theorems \[Th1\], \[Th2\], and \[Th3\].

As a remark, due to the scaling law , every extremal function of or generates in fact an infinite family of extremal functions.

Preliminary to the proof of Theorem \[Th5\], we prove the following result.

\[Pr1\] Let $\overrightarrow{\theta}={\left(}\theta_1,\dotsc,\theta_n{\right)}$ be such that $\theta_i>0$ for all $i=1,\dotsc,n$ and $\sum_{i=1}^n1/\theta_i=n/p$. Then the following assertions hold true.

1.  For any extremal function $u$ of , $u\circ\tau_{\overrightarrow{\theta}}$ is an extremal function of , where $$\label{Pr1Eq1}
    \tau_{\overrightarrow{\theta}}{\left(}x{\right)}:=\left(\lambda_{\overrightarrow{\theta},1}x_1,\dotsc,\lambda_{\overrightarrow{\theta},n}x_n\right),\quad\lambda_{\overrightarrow{\theta},i}:=\theta_i^{1/\theta_i}\prod_{j=1}^ n\theta_j^{-p/{\left(}n\theta_i\theta_j{\right)}}\,,$$ for all $x\in{\mathbb{R}}^n$ and $i=1,\dotsc,n$.

2.  For any extremal function $u$ of , $u\circ\sigma_{\overrightarrow{\theta},u}\circ\tau_{\overrightarrow{\theta}}^{-1}$ is an extremal function of , where $\tau_{\overrightarrow{\theta}}$ is as in and $$\label{Pr1Eq2}
    \sigma_{\overrightarrow{\theta},u}{\left(}x{\right)}:=\left(\mu_{\overrightarrow{\theta},1}{\left(}u{\right)}x_1,\dotsc,\mu_{\overrightarrow{\theta},n}{\left(}u{\right)}x_n\right),\quad\mu_{\overrightarrow{\theta},i}{\left(}u{\right)}:=\frac{\overset{n}{\underset{j=1}{\prod}}{\left(}\int_{{\mathbb{R}}^n}\left|\partial_{x_j}u\right|^{p_j}dx{\right)}^{p/{\left(}n\theta_ip_j{\right)}}}{{\left(}\int_{{\mathbb{R}}^n}\left|\partial_{x_i}u\right|^{p_i}dx{\right)}^{1/p_i}}\,,$$ for all $x\in{\mathbb{R}}^n$ and $i=1,\dotsc,n$.

We begin with proving Point (i). We fix an extremal function $u_0$ of . Since $\sum_{i=1}^n1/\theta_i=n/p$, we obtain $$\label{Pr1Eq3}
\prod_{i=1}^n{\left(}\int_{{\mathbb{R}}^n}\big|\partial_{x_i}\big(u_0\circ\tau_{\overrightarrow{\theta}}\big)\big|^{p_i}dx{\right)}^{p/{\left(}np_i{\right)}}\le\frac{p}{n}\sum_{i=1}^n\frac{1}{\theta_i}{\left(}\int_{{\mathbb{R}}^n}\big|\partial_{x_i}\big(u_0\circ\tau_{\overrightarrow{\theta}}\big)\big|^{p_i}dx{\right)}^{\theta_i/p_i}.$$ For any function $u\in D^{1,\operatorname{\overrightarrow{p}}}{\left(}{\mathbb{R}}^n{\right)}$, simple calculations give $$\label{Pr1Eq4}
\sum_{i=1}^n\frac{1}{\theta_i}{\left(}\int_{{\mathbb{R}}^n}\big|\partial_{x_i}\big(u\circ\tau_{\overrightarrow{\theta}}\big)\big|^{p_i}dx{\right)}^{\theta_i/p_i}={\left(}\prod_{j=1}^ n\theta_j^{-p/{\left(}n\theta_j{\right)}}{\right)}\sum_{i=1}^n{\left(}\int_{{\mathbb{R}}^n}\left|\partial_{x_i}u\right|^{p_i}dx{\right)}^{\theta_i/p_i}$$ and $$\label{Pr1Eq5}
\int_{{\mathbb{R}}^n}\big|u\circ\tau_{\overrightarrow{\theta}}\big|^{p^*}dx=\int_{{\mathbb{R}}^n}\left|u\right|^{p^*}dx\,.$$ By invertibility of $\tau_{\overrightarrow{\theta}}$ and since $u_0$ is an extremal function of , it follows from and that $$\label{Pr1Eq6}
\frac{\overset{n}{\underset{i=1}{\sum}}\frac{1}{\theta_i}{\left(}\int_{{\mathbb{R}}^n}\big|\partial_{x_i}\big(u_0\circ\tau_{\overrightarrow{\theta}}\big)\big|^{p_i}dx{\right)}^{\theta_i/p_i}}{\big(\int_{{\mathbb{R}}^n}\big|u_0\circ\tau_{\overrightarrow{\theta}}\big|^{p^*}dx\big)^{p/p^*}}=\inf_{\underset{u\ne0}{u\in D^{1,\operatorname{\overrightarrow{p}}}{\left(}{\mathbb{R}}^n{\right)}}}\frac{\overset{n}{\underset{i=1}{\sum}}\frac{1}{\theta_i}{\left(}\int_{{\mathbb{R}}^n}\left|\partial_{x_i}u\right|^{p_i}dx{\right)}^{\theta_i/p_i}}{\left(\int_{{\mathbb{R}}^n}\left|u\right|^{p^*}dx\right)^{p/p^*}}\,.$$ Now, we claim that $$\label{Pr1Eq7}
\inf_{\underset{u\ne0}{u\in D^{1,\operatorname{\overrightarrow{p}}}{\left(}{\mathbb{R}}^n{\right)}}}\frac{\overset{n}{\underset{i=1}{\sum}}\frac{1}{\theta_i}{\left(}\int_{{\mathbb{R}}^n}\left|\partial_{x_i}u\right|^{p_i}dx{\right)}^{\theta_i/p_i}}{\left(\int_{{\mathbb{R}}^n}\left|u\right|^{p^*}dx\right)^{p/p^*}}\le\frac{n}{p}\cdot\inf_{\underset{u\ne0}{u\in D^{1,\operatorname{\overrightarrow{p}}}{\left(}{\mathbb{R}}^n{\right)}}}\frac{\overset{n}{\underset{i=1}{\prod}}{\left(}\int_{{\mathbb{R}}^n}\left|\partial_{x_i}u\right|^{p_i}dx{\right)}^{p/{\left(}np_i{\right)}}}{\left(\int_{{\mathbb{R}}^n}\left|u\right|^{p^*}dx\right)^{p/p^*}}\,.$$ We prove this claim. For any function $u\in D^{1,\operatorname{\overrightarrow{p}}}{\left(}{\mathbb{R}}^n{\right)}$, $u\ne0$, by applying the change of scale , we obtain $$\label{Pr1Eq8}
{\left(}\int_{{\mathbb{R}}^n}\big|\partial_{x_i}\big(u\circ\sigma_{\overrightarrow{\theta},u}\big)\big|^{p_i}dx{\right)}^{\theta_i/p_i}=\prod_{j=1}^n{\left(}\int_{{\mathbb{R}}^n}\left|\partial_{x_j}u\right|^{p_j}dx{\right)}^{p/{\left(}np_j{\right)}}$$ for all $i=1,\dotsc,n$, and $$\label{Pr1Eq9}
\int_{{\mathbb{R}}^n}\big|u\circ\sigma_{\overrightarrow{\theta},u}\big|^{p^*}dx=\int_{{\mathbb{R}}^n}\left|u\right|^{p^*}dx\,.$$ Since $\sum_{i=1}^n1/\theta_i=n/p$, it follows from and that $$\label{Pr1Eq10}
\frac{\overset{n}{\underset{i=1}{\sum}}\frac{1}{\theta_i}{\left(}\int_{{\mathbb{R}}^n}\big|\partial_{x_i}\big(u\circ\sigma_{\overrightarrow{\theta},u}\big)\big|^{p_i}dx{\right)}^{\theta_i/p_i}}{\left(\int_{{\mathbb{R}}^n}\big|u\circ\sigma_{\overrightarrow{\theta},u}\big|^{p^*}dx\right)^{p/p^*}}=\frac{n}{p}\cdot\frac{\overset{n}{\underset{i=1}{\prod}}{\left(}\int_{{\mathbb{R}}^n}\left|\partial_{x_i}u\right|^{p_i}dx{\right)}^{p/{\left(}np_i{\right)}}}{\left(\int_{{\mathbb{R}}^n}\left|u\right|^{p^*}dx\right)^{p/p^*}}\,,$$ and hence we obtain . It follows from , and that $u_0\circ\tau_{\overrightarrow{\theta}}$ is an extremal function of . This ends the proof of Point (i).

Now, we prove Point (ii). We fix an extremal function $u_0$ of . By and since $\sum_{i=1}^n1/\theta_i=n/p$, we obtain $$\begin{aligned}
\label{Pr1Eq11}
\frac{\overset{n}{\underset{i=1}{\sum}}\frac{1}{\theta_i}{\left(}\int_{{\mathbb{R}}^n}\big|\partial_{x_i}\big(u_0\circ\sigma_{\overrightarrow{\theta},u_0}\big)\big|^{p_i}dx{\right)}^{\theta_i/p_i}}{\left(\int_{{\mathbb{R}}^n}\big|u_0\circ\sigma_{\overrightarrow{\theta},u_0}\big|^{p^*}dx\right)^{p/p^*}}&=\frac{n}{p}\cdot\inf_{\underset{u\ne0}{u\in D^{1,\operatorname{\overrightarrow{p}}}{\left(}{\mathbb{R}}^n{\right)}}}\frac{\overset{n}{\underset{i=1}{\prod}}{\left(}\int_{{\mathbb{R}}^n}\left|\partial_{x_i}u\right|^{p_i}dx{\right)}^{p/{\left(}np_i{\right)}}}{\left(\int_{{\mathbb{R}}^n}\left|u\right|^{p^*}dx\right)^{p/p^*}}\nonumber\\
&\le\inf_{\underset{u\ne0}{u\in D^{1,\operatorname{\overrightarrow{p}}}{\left(}{\mathbb{R}}^n{\right)}}}\frac{\overset{n}{\underset{i=1}{\sum}}\frac{1}{\theta_i}{\left(}\int_{{\mathbb{R}}^n}\left|\partial_{x_i}u\right|^{p_i}dx{\right)}^{\theta_i/p_i}}{\left(\int_{{\mathbb{R}}^n}\left|u\right|^{p^*}dx\right)^{p/p^*}}\,.\end{aligned}$$ It follows from and that $u_0\circ\sigma_{\overrightarrow{\theta},u_0}\circ\tau_{\overrightarrow{\theta}}^{-1}$ is an extremal function of . This ends the proof of Point (ii).

Now, we can prove Theorem \[Th5\] by using Proposition \[Pr1\].

We prove the results for the sole inequality . The results for and then follow from Proposition \[Pr1\].

First, in case $p_+<p^*$, the existence of extremal functions of follows from the work of El Hamidi–Rakotoson [@ElHRak]. Indeed, it has been proven in [@ElHRak] that there exist minimizers for $$\label{Th5Eq1}
\mathcal{I}:=\inf_{\underset{\int_{{\mathbb{R}}^n}\left|u\right|^{p^*}dx=1}{u\in D^{1,\operatorname{\overrightarrow{p}}}{\left(}{\mathbb{R}}^n{\right)}}}\sum_{i=1}^n\frac{1}{p_i}\int_{{\mathbb{R}}^n}\left|\partial_{x_i}u\right|^{p_i}dx\,.$$ This infimum is connected with by the change of scale $u\mapsto\mu_{\operatorname{\overrightarrow{p}},u}^{-1}\cdot u\circ\rho_{\operatorname{\overrightarrow{p}},u}$, where $$\rho_{\operatorname{\overrightarrow{p}},u}{\left(}x{\right)}:=\mu_{\operatorname{\overrightarrow{p}},u}\cdot\tau_{\operatorname{\overrightarrow{p}}}{\left(}x{\right)},\quad\mu_{\operatorname{\overrightarrow{p}},u}:={\left(}\int_{{\mathbb{R}}^n}\left|u\right|^{p^*}dx{\right)}^{p/{\left(}np^*{\right)}},$$ and $\tau_{\operatorname{\overrightarrow{p}}}{\left(}x{\right)}$ is as in for all $x\in{\mathbb{R}}^n$ and $u\in D^{1,\operatorname{\overrightarrow{p}}}{\left(}{\mathbb{R}}^n{\right)}$, $u\ne0$. More precisely, simple calculations give $$\label{Th5Eq2}
\sum_{i=1}^n\frac{1}{p_i}\int_{{\mathbb{R}}^n}\big|\partial_{x_i}\big(\mu_{\operatorname{\overrightarrow{p}},u}^{-1}\cdot u\circ\rho_{\operatorname{\overrightarrow{p}},u}\big)\big|^{p_i}dx={\left(}\prod_{j=1}^np_j^{-p/{\left(}np_j{\right)}}{\right)}\frac{\overset{n}{\underset{i=1}{\sum}}\int_{{\mathbb{R}}^n}\big|\partial_{x_i}u\big|^{p_i}dx}{{\left(}\int_{{\mathbb{R}}^n}\left|u\right|^{p^*}dx{\right)}^{p/p^*}}$$ and $$\label{Th5Eq3}
\int_{{\mathbb{R}}^n}\big|\mu_{\operatorname{\overrightarrow{p}},u}^{-1}\cdot u\circ\rho_{\operatorname{\overrightarrow{p}},u}\big|^{p^*}dx=1\,,$$ and hence $$\label{Th5Eq4}
\mathcal{I}\le{\left(}\prod_{j=1}^np_j^{-p/{\left(}np_j{\right)}}{\right)}\inf_{\underset{u\ne0}{u\in D^{1,\operatorname{\overrightarrow{p}}}{\left(}{\mathbb{R}}^n{\right)}}}\frac{\overset{n}{\underset{i=1}{\sum}}\int_{{\mathbb{R}}^n}\big|\partial_{x_i}u\big|^{p_i}dx}{{\left(}\int_{{\mathbb{R}}^n}\left|u\right|^{p^*}dx{\right)}^{p/p^*}}\,.$$ In particular, for any minimizer $u$ of , since $\mu_{\operatorname{\overrightarrow{p}},u}=1$ and $\rho_{\operatorname{\overrightarrow{p}},u}=\tau_{\operatorname{\overrightarrow{p}}}$, it follows from – that $u\circ\tau_{\operatorname{\overrightarrow{p}}}^{-1}$ is an extremal function of .

Next, we prove that the extremal functions of do not change sign. We let $C_0$ be the best constant and $u$ be an extremal function of . By writing $u=u_+-u_-$, where $u_+:=\max{\left(}u,0{\right)}$ and $u_-:=\max{\left(}-u,0{\right)}$, we obtain $$\begin{gathered}
\label{Th5Eq5}
\sum_{i=1}^n\int_{{\mathbb{R}}^n}\left|\partial_{x_i}u\right|^{p_i}dx={\left(}\frac{1}{C_0}\int_{{\mathbb{R}}^n}\left|u\right|^{p^*}dx{\right)}^{p/p_*}={\left(}\frac{1}{C_0}\int_{{\mathbb{R}}^n}u_-^{p^*}dx+\frac{1}{C_0}\int_{{\mathbb{R}}^n}u_+^{p^*}dx{\right)}^{p/p_*}\\
\le{\left(}{\left(}\sum_{i=1}^n\int_{{\mathbb{R}}^n}\left|\partial_{x_i}u_-\right|^{p_i}dx{\right)}^{p^*/p}+{\left(}\sum_{i=1}^n\int_{{\mathbb{R}}^n}\left|\partial_{x_i}u_+\right|^{p_i}dx{\right)}^{p^*/p}{\right)}^{p/p_*}.\end{gathered}$$ It follows from that either $u_-=0$ or $u_+=0$, and hence we obtain that the function $u$ has constant sign.

Finally, from the Euler-Lagrange equation satisfied by $u$, namely $$-\sum_{i=1}^np_i\partial_{x_i}{\left(}\left|\partial_{x_i}u\right|^{p_i-2}\partial_{x_i}u{\right)}=\lambda{\left(}u{\right)}\left|u\right|^{p^*-2}u,\quad\text{where}\quad\lambda{\left(}u{\right)}:=\frac{\overset{n}{\underset{i=1}{\sum}}p_i\int_{{\mathbb{R}}^n}\big|\partial_{x_i}u\big|^{p_i}dx}{\int_{{\mathbb{R}}^n}\left|u\right|^{p^*}dx}\,,$$ we derive that the function $x\in{\mathbb{R}}^n\mapsto\mu u{\left(}\mu_1x_1,\dotsc,\mu_nx_n{\right)}$ with $\mu_i:={\left(}\lambda{\left(}u{\right)}/p_i{\right)}^{1/p_i}$ for all $i=1,\dotsc,n$ is a solution of with $f{\left(}x,u{\right)}=\left|u\right|^{p^*-2}u$. This ends the proof of Theorem \[Th5\].

Preliminary results {#Sec3}
===================

From now on, we are concerned with the general case of an arbitrary solution of .

For any $s\in{\left(}0,\infty{\right)}$ and any domain $\Omega\subset{\mathbb{R}}^n$, we define the weak Lebesgue space $L^{s,\infty}{\left(}\Omega{\right)}$ as the set of all measurable functions $u:\Omega\to{\mathbb{R}}$ such that $$\left\|u\right\|_{L^{s,\infty}{\left(}\Omega{\right)}}:=\sup_{h>0}\big(h\cdot\operatorname{meas}{\left(}\left\{\left|u\right|>h\right\}{\right)}^{1/s}\big)<\infty\,,$$ where $\operatorname{meas}{\left(}\left\{\left|u\right|>h\right\}{\right)}$ is the measure of the set $\left\{x\in\Omega:\,\left|u{\left(}x{\right)}\right|>h\right\}$. The map $\left\|\cdot\right\|_{L^{s,\infty}{\left(}\Omega{\right)}}$ defines a quasi-norm on $L^{s,\infty}{\left(}\Omega{\right)}$. We refer, for instance, to the book of Grafakos [@Gra] for the material on weak Lebesgue spaces.

The first result in this section is as follows.

\[Lem1\] Assume that $p_+<p^*$. Let $f:{\mathbb{R}}^n\times{\mathbb{R}}\to{\mathbb{R}}$ be a Caratheodory function such that holds true. Then any solution of belongs to $W^{1,\infty}{\left(}{\mathbb{R}}^n{\right)}\cap L^{p_*-1,\infty}{\left(}{\mathbb{R}}^n{\right)}$, and hence by interpolation, to $L^s{\left(}{\mathbb{R}}^n{\right)}$ for all $s\in{\left(}p_*-1,\infty{\right]}$.

The $L^\infty$–boundedness of the solutions follows from a straightforward adaptation of El Hamidi–Rakotoson [@ElHRak]\*[Propositions 1 and 2]{}, the first proposition being in turn adapted from Fragalà–Gazzola–Kawohl [@FraGazKaw]\*[Theorem 2]{}.

Once we have the $L^\infty$–boundedness of the solutions, we obtain the $L^\infty$–boundedness of the derivatives by applying Lieberman’s gradient estimates [@Lie].

The proof of the $L^{p_*-1,\infty}$–boundedness of the solutions follows exactly the same arguments as in Vétois [@Vet3]\*[Lemma 2.2]{}. One only has to replace $\left|\nabla u\right|^p$ by $\sum_{i=1}^n\left|\partial_{x_i}u\right|^{p_i}$.

For any solution $u$ of , by Proposition \[Pr2\] in Appendix \[App\], we obtain $$\label{Eq10}
-\Delta_{\operatorname{\overrightarrow{p}}}\left|u\right|\le f{\left(}x,u{\right)}\cdot\operatorname{sgn}{\left(}u{\right)}\le\Lambda\left|u\right|^{p^*-1}\quad\text{in }{\mathbb{R}}^n,$$ where $\operatorname{sgn}{\left(}u{\right)}$ denotes the sign of $u$ and the inequality is in the sense that for any nonnegative, smooth function $\varphi$ with compact support in ${\mathbb{R}}^n$, we have $$\sum_{i=1}^n\int_{{\mathbb{R}}^n}\left|\partial_{x_i}\left|u\right|\right|^{p_i-2}{\left(}\partial_{x_i}\left|u\right|{\right)}{\left(}\partial_{x_i}\varphi{\right)}dx\le\Lambda\int_{{\mathbb{R}}^n}\left|u\right|^{p^*-1}\varphi\,dx\,.$$

We prove the following result.

\[Lem2\] For any real number $\Lambda>0$ and any nonnegative, nontrivial solution $v\in D^{1,p}{\left(}{\mathbb{R}}^n{\right)}$ of the inequality $-\Delta_{\operatorname{\overrightarrow{p}}}v\le\Lambda v^{p^*-1}$ in ${\mathbb{R}}^n$, we have $\left\|v\right\|_{L^{p^*}{\left(}{\mathbb{R}}^n{\right)}}\ge\kappa_0$ for some constant $\kappa_0=\kappa_0{\left(}n,p,\Lambda{\right)}>0$.

By testing the inequality $-\Delta_{\operatorname{\overrightarrow{p}}}v\le\Lambda v^{p^*-1}$ with the function $v$, and applying the anisotropic Sobolev inequality, we obtain $$\label{Lem2Eq}
\Lambda\int_{{\mathbb{R}}^n}v^{p^*}dx\ge\sum_{i=1}^n\int_{{\mathbb{R}}^n}\left|\partial_{x_i}v\right|^{p_i}dx\ge K{\left(}\int_{{\mathbb{R}}^n}v^{p^*}dx{\right)}^{\frac{n-p}{n}}$$ for some constant $K=K{\left(}n,\operatorname{\overrightarrow{p}}{\right)}$. The result then follows from with $\kappa_0:={\left(}K/\Lambda{\right)}^{\frac{n-p}{p^2}}$.

As a last result in this section, we prove the following decay estimate. This result is not sharp, but it turns out to be a crucial ingredient in what follows.

\[Lem3\] Assume that $p_+<p^*$. Let $\kappa_0$ be as in Lemma \[Lem2\], $f:{\mathbb{R}}^n\times{\mathbb{R}}\to{\mathbb{R}}$ be a Caratheodory function such that holds true, and $u$ be a solution of . For any $\kappa>0$, we define $$\label{Lem3Eq1}
r_\kappa{\left(}u{\right)}:=\inf\big(\big\{r>0\,:\,\left\|u\right\|_{L^{p^*}{\left(}{\mathbb{R}}^n\backslash B_{\operatorname{\overrightarrow{p}}}{\left(}0,r{\right)}{\right)}}<\kappa\big\}\big),$$ where $B_{\operatorname{\overrightarrow{p}}}{\left(}0,r{\right)}$ is the open ball of center 0 and radius $r$ with respect to the distance function $d_{\operatorname{\overrightarrow{p}}}$ defined as $$\label{Lem3Eq2}
d_{\operatorname{\overrightarrow{p}}}{\left(}x,y{\right)}:=\sum_{i=1}^n\left|x_i-y_i\right|^{\frac{\delta p_i}{p^*-p_i}}\quad\text{with}\quad\delta:=\frac{p^*-p_+}{p_+},$$ for all $x,y\in{\mathbb{R}}^n$. Then for any $\kappa\in{\left(}0,\kappa_0{\right)}$ and $r>r_\kappa{\left(}u{\right)}$, there exists a constant $K_0=K_0\big(n,\operatorname{\overrightarrow{p}},\Lambda,\kappa,r,r_\kappa{\left(}u{\right)},\left\|u\right\|_{L^{p^*}{\left(}{\mathbb{R}}^n{\right)}}\big)$ such that $$\label{Lem3Eq3}
\left|u{\left(}x{\right)}\right|\le K_0\bigg(\sum_{i=1}^n\left|x_i\right|^{\frac{p_i}{p^*-p_i}}\bigg)^{-1}\quad\text{for all }x\in{\mathbb{R}}^n\backslash B_{\operatorname{\overrightarrow{p}}}{\left(}0,r{\right)}.$$

This proof is adapted from Vétois [@Vet3]\*[Lemma 3.1]{} We fix $\Lambda>0$, $\kappa\in{\left(}0,\kappa_0{\right)}$, $\kappa'>\kappa_0$, $r>0$, and $r'\in{\left(}0,r{\right)}$. We claim that in order to obtain Lemma \[Lem3\], it is sufficient to prove that there exists a constant $K_1=K_1{\left(}n,\operatorname{\overrightarrow{p}},\kappa,\kappa',r,r'{\right)}$ such that for any solution $u$ of such that $r_\kappa{\left(}u{\right)}\le r'$ and $\left\|u\right\|_{L^{p^*}{\left(}{\mathbb{R}}^n{\right)}}\le\kappa'$, we have $$\label{Lem3Eq4}
d_{\operatorname{\overrightarrow{p}}}{\left(}x,B_{\operatorname{\overrightarrow{p}}}{\left(}0,r''{\right)}{\right)}\left|u{\left(}x{\right)}\right|^\delta\le K_1\quad\text{for all }x\in{\mathbb{R}}^n\backslash B_{\operatorname{\overrightarrow{p}}}{\left(}0,r{\right)},$$ where $r'':={\left(}r+r'{\right)}/2$. Indeed, for any $x\in{\mathbb{R}}^n\backslash B_{\operatorname{\overrightarrow{p}}}{\left(}0,r{\right)}$, we can write $$\label{Lem3Eq5}
d_{\operatorname{\overrightarrow{p}}}{\left(}x,0{\right)}\le d_{\operatorname{\overrightarrow{p}}}{\left(}x,B_{\operatorname{\overrightarrow{p}}}{\left(}0,r''{\right)}{\right)}+r''\le d_{\operatorname{\overrightarrow{p}}}{\left(}x,B_{\operatorname{\overrightarrow{p}}}{\left(}0,r''{\right)}{\right)}+\frac{r''}{r}d_{\operatorname{\overrightarrow{p}}}{\left(}x,0{\right)},$$ and hence by putting together and , we obtain $$\label{Lem3Eq6}
d_{\operatorname{\overrightarrow{p}}}{\left(}x,0{\right)}\left|u{\left(}x{\right)}\right|^\delta\le\frac{r}{r-r''}\cdot K_1=\frac{2r}{r-r'}\cdot K_1.$$ By definition of $d_{\operatorname{\overrightarrow{p}}}$, then follows from . This proves our claim.

We prove by contradiction. Suppose that for any $\alpha\in{\mathbb{N}}$, there exists a Caratheodory function $f_\alpha:{\mathbb{R}}^n\times{\mathbb{R}}\to{\mathbb{R}}$ such that holds true, a solution $u_\alpha$ of with $f=f_\alpha$ such that $r_\kappa{\left(}u_\alpha{\right)}\le r'$ and $\left\|u_\alpha\right\|_{L^{p^*}{\left(}{\mathbb{R}}^n{\right)}}\le\kappa'$, and a point $x_\alpha\in{\mathbb{R}}^n\backslash B{\left(}0,r{\right)}$ such that $$\label{Lem3Eq7}
d_{\operatorname{\overrightarrow{p}}}{\left(}x_\alpha,B_{\operatorname{\overrightarrow{p}}}{\left(}0,r''{\right)}{\right)}\left|u_\alpha{\left(}x_\alpha{\right)}\right|^\delta>2\alpha\,.$$ It follows from and Poláčik–Quittner–Souplet [@PolQuiSou]\*[Lemma 5.1]{} that there exists $y_\alpha\in{\mathbb{R}}^n\backslash B_{\operatorname{\overrightarrow{p}}}{\left(}0,r''{\right)}$ such that $$\label{Lem3Eq8}
d_{\operatorname{\overrightarrow{p}}}{\left(}y_\alpha,B_{\operatorname{\overrightarrow{p}}}{\left(}0,r''{\right)}{\right)}\left|u_\alpha{\left(}y_\alpha{\right)}\right|^\delta>2\alpha\,,\quad\left|u_\alpha{\left(}x_\alpha{\right)}\right|\le\left|u_\alpha{\left(}y_\alpha{\right)}\right|,$$ and $$\label{Lem3Eq9}
\left|u_\alpha{\left(}y{\right)}\right|\le2^{1/\delta}\left|u_\alpha{\left(}y_\alpha{\right)}\right|\quad\text{for all }y\in B_{\operatorname{\overrightarrow{p}}}\big(y_\alpha,\alpha\,\left|u_\alpha{\left(}y_\alpha{\right)}\right|^{-\delta}\big).$$ For any $\alpha$ and $y\in{\mathbb{R}}^n$, we define $$\label{Lem3Eq10}
\widetilde{u}_\alpha{\left(}y{\right)}:=\left|u_\alpha{\left(}y_\alpha{\right)}\right|^{-1}\cdot u_\alpha{\left(}\tau_\alpha{\left(}y{\right)}{\right)},$$ where $$\tau_\alpha{\left(}y{\right)}:=y_\alpha+\big(\left|u_\alpha{\left(}y_\alpha{\right)}\right|^{\frac{p_1-p^*}{p_1}}y_1,\dots,\left|u_\alpha{\left(}y_\alpha{\right)}\right|^{\frac{p_n-p^*}{p_n}}y_n\big).$$ It follows from and that $$\label{Lem3Eq11}
\left|\widetilde{u}_\alpha{\left(}0{\right)}\right|=1\quad\text{and}\quad\left|\widetilde{u}_\alpha{\left(}y{\right)}\right|\le2^{1/\delta}\quad\text{for all }y\in B_{\operatorname{\overrightarrow{p}}}{\left(}0,\alpha{\right)}.$$ Moreover, by , we obtain $$\label{Lem3Eq12}
-\Delta_{\operatorname{\overrightarrow{p}}}\widetilde{u}_\alpha=\left|u_\alpha{\left(}y_\alpha{\right)}\right|^{1-p^*}\cdot f_\alpha{\left(}\tau_\alpha{\left(}y{\right)},\left|u_\alpha{\left(}y_\alpha{\right)}\right|\cdot\widetilde{u}_\alpha{\right)}\quad\text{in }{\mathbb{R}}^n,$$ and gives $$\label{Lem3Eq13}
\left|u_\alpha{\left(}y_\alpha{\right)}\right|^{1-p^*}\cdot\left|f_\alpha{\left(}\tau_\alpha{\left(}y{\right)},\left|u_\alpha{\left(}y_\alpha{\right)}\right|\cdot\widetilde{u}_\alpha{\right)}\right|\le\Lambda\left|\widetilde{u}_\alpha\right|^{p^*-1}.$$ By Lieberman’s gradient estimates [@Lie], it follows from and that there exists a constant $C>0$ such that for any $R>0$, we have $$\label{Lem3Eq14}
\left\|\nabla\widetilde{u}_\alpha\right\|_{L^\infty(B_{\operatorname{\overrightarrow{p}}}{\left(}0,R{\right)})}\le C$$ for large $\alpha$. By Arzela–Ascoli Theorem and a diagonal argument, it follows from and that ${\left(}\widetilde{u}_\alpha{\right)}_\alpha$ converges up to a subsequence in $C^0_{\operatorname{loc}}{\left(}{\mathbb{R}}^n{\right)}$ to some Lipschitz continuous function $\widetilde{u}_\infty$ such that $\left|\widetilde{u}_\infty{\left(}0{\right)}\right|=1$. Moreover, by testing – with $\widetilde{u}_\alpha$, we obtain $$\label{Lem3Eq15}
\sum_{i=1}^n\int_{{\mathbb{R}}^n}\left|\partial_{x_i}\widetilde{u}_\alpha\right|^{p_i}dx\le\Lambda\int_{{\mathbb{R}}^n}\left|\widetilde{u}_\alpha\right|^{p^*}dx=\Lambda\int_{{\mathbb{R}}^n}\left|u_\alpha\right|^{p^*}dx\le\Lambda{\left(}\kappa'{\right)}^{p^*}.$$ Since $\left|\partial_{x_i}\left|\widetilde{u}_\alpha\right|\right|=\left|\partial_{x_i}\widetilde{u}_\alpha\right|$ a.e. in ${\mathbb{R}}^n$, it follows from that ${\left(}\left|\widetilde{u}_\alpha\right|{\right)}_\alpha$ converges weakly up to a subsequence to $\left|\widetilde{u}_\infty\right|$ in $D^{1,\operatorname{\overrightarrow{p}}}{\left(}{\mathbb{R}}^n{\right)}$. Passing to the limit into –, we then obtain that $\left|\widetilde{u}_\infty\right|$ is a weak solution of the inequality $$\label{Lem3Eq16}
-\Delta_{\operatorname{\overrightarrow{p}}}\left|\widetilde{u}_\infty\right|\le\Lambda\left|\widetilde{u}_\infty\right|^{p^*-1}\quad\text{in }{\mathbb{R}}^n.$$ In particular, since $\left|\widetilde{u}_\infty{\left(}0{\right)}\right|=1$, it follows from Lemma \[Lem2\] that $\left\|u_\infty\right\|_{L^{p^*}{\left(}{\mathbb{R}}^n{\right)}}\ge\kappa_0$, and hence there exists a real number $R>0$ such that $$\label{Lem3Eq17}
\left\|u_\infty\right\|_{L^{p^*}{\left(}B{\left(}0,R{\right)}{\right)}}>\kappa\,.$$ On the other hand, we have $$\label{Lem3Eq18}
\left\|\widetilde{u}_\alpha\right\|_{L^{p^*}(B_{\operatorname{\overrightarrow{p}}}{\left(}0,R{\right)})}=\left\|u_\alpha\right\|_{L^{p^*}{\left(}B_{\operatorname{\overrightarrow{p}}}{\left(}y_\alpha,R\cdot\left|u_\alpha{\left(}y_\alpha{\right)}\right|^{-\delta}{\right)}{\right)}}.$$ By and since $r_\kappa{\left(}u_\alpha{\right)}<r''$, we obtain $$\label{Lem3Eq19}
B_{\operatorname{\overrightarrow{p}}}\big(y_\alpha,R\cdot\left|u_\alpha{\left(}y_\alpha{\right)}\right|^{-\delta}\big)\cap B_{\operatorname{\overrightarrow{p}}}{\left(}0,r_\kappa{\left(}u_\alpha{\right)}{\right)}=\emptyset$$ for large $\alpha$. By definition of $r_\kappa{\left(}u_\alpha{\right)}$, it follows from and that $$\label{Lem3Eq20}
\left\|\widetilde{u}_\alpha\right\|_{L^{p^*}(B_{\operatorname{\overrightarrow{p}}}{\left(}0,R{\right)})}\le\kappa$$ for large $\alpha$, which is in contradiction with . This ends the proof of Lemma \[Lem3\].

The reverse Hölder-type inequalities {#Sec4}
====================================

The following result is a key step in the Moser-type iteration scheme that we develop in the next section.

\[Lem4\] Assume that $p_+<p^*$. Let $f:{\mathbb{R}}^n\times{\mathbb{R}}\to{\mathbb{R}}$ be a Caratheodory function such that holds true, $u$ be a solution of , and $\kappa$, $r$, and $K_0$ be as in Lemma \[Lem3\]. Let $\mathcal{I}_1$ and $\mathcal{I}_2$ be two disjoint subsets of $\left\{1,\dotsc,n\right\}$, $\mathcal{I}_2\ne\emptyset$, and ${\overrightarrow{q}}={\left(}q_i{\right)}_{i\in\mathcal{I}_1\cup\mathcal{I}_2}$ be such that $q_i>1$ for all $i\in\mathcal{I}_1\cup\mathcal{I}_2$. Then there exists a constant $c_0=c_0{\left(}n,\operatorname{\overrightarrow{p}},\Lambda,K_0{\right)}>1$ such that for any $R_1,R_2>0$, $\lambda<\lambda'\in{\left(}0,1/2{\right]}$, and $\gamma>p_*-1$ such that $\Omega_{{\overrightarrow{q}}}{\left(}\mathcal{I}_1,R_1,\mathcal{I}_2,R_2,\lambda'{\right)}\cap B_{\operatorname{\overrightarrow{p}}}{\left(}0,\max{\left(}r,1{\right)}{\right)}=\emptyset$ and $\Omega_{{\overrightarrow{q}}}{\left(}\mathcal{I}_1,R_1,\mathcal{I}_2,R_2,\lambda'{\right)}\cap\operatorname{supp}{\left(}u{\right)}$ is bounded, where $\Omega_{{\overrightarrow{q}}}{\left(}\mathcal{I}_1,R_1,\mathcal{I}_2,R_2,\lambda{\right)}$ is as in , we have $$\begin{gathered}
\label{Lem4Eq1}
\left\|u\right\|_{L^\gamma{\left(}\Omega_{{\overrightarrow{q}}}{\left(}\mathcal{I}_1,R_1,\mathcal{I}_2,R_2,\lambda{\right)}{\right)}}^\gamma\le c_0\gamma^{p^*}\max_{i\in\mathcal{I}_1\cup\mathcal{I}_2}\Big(\min{\left(}1,\gamma-p_*+1{\right)}^{-p_i}{\left(}\lambda'-\lambda{\right)}^{-p_i}q_i^{p_i}\\
\times R_{\delta_i}^{-\frac{p_i}{q_i}}\left\|u\right\|_{L^{\gamma_i}{\left(}\Omega_{{\overrightarrow{q}}}{\left(}\mathcal{I}_1,R_1,\mathcal{I}_2,R_2,\lambda'{\right)}{\right)}}^{\gamma_i}\Big)^\frac{n}{n-p},\end{gathered}$$ where $\delta_i:=1$ if $i\in\mathcal{I}_1$, $\delta_i:=2$ if $i\in\mathcal{I}_2$, and $\gamma_i:=\frac{n-p}{n}\gamma+p_i-p$ for all $i\in\mathcal{I}_1\cup\mathcal{I}_2$.

Preliminary to the proof of Lemma \[Lem4\], we prove the following result.

\[Lem5\] Let $v$ be a nonnegative solution in $D^{1,\operatorname{\overrightarrow{p}}}{\left(}{\mathbb{R}}^n{\right)}$ of $$\label{Lem5Eq1}
-\Delta_{\operatorname{\overrightarrow{p}}}v\le\Lambda v^{p^*-1}\quad\text{in }{\mathbb{R}}^n,$$ for some real number $\Lambda>0$, where the inequality must be understood in the weak sense as in . Let $\beta>-1$ and $\eta\in C^1{\left(}{\mathbb{R}}^n{\right)}$ be such that $0\le\eta\le1$ in ${\mathbb{R}}^n$, $\eta v$ has compact support, and $\eta^{{\left(}\beta+p_-{\right)}/p_+}\in C^1{\left(}{\mathbb{R}}^n{\right)}$, where $p_-:=\min{\left(}\left\{p_i\in\operatorname{\overrightarrow{p}}\right\}{\right)}$ and $p_+:=\max{\left(}\left\{p_i\in\operatorname{\overrightarrow{p}}\right\}{\right)}$. Then there exists a constant $C=C{\left(}n,\operatorname{\overrightarrow{p}}{\right)}$ such that $$\begin{gathered}
\label{Lem5Eq2}
{\left(}\int_{{\mathbb{R}}^n}{\left(}\eta v{\right)}^{\frac{n{\left(}\beta+p{\right)}}{n-p}}dx{\right)}^{\frac{n-p}{n}}\le C{\left(}\left|\beta\right|^p+1{\right)}\bigg(\Lambda{\left(}\beta+1{\right)}^{-1}\int_{{\mathbb{R}}^n}\eta^{\beta+p_-}v^{\beta+p^*}dx\\
+\sum_{i=1}^n\min{\left(}1,\beta+1{\right)}^{-p_i}\int_{{\mathbb{R}}^n}\left|\partial_{x_i}\eta\right|^{p_i}\eta^{\beta+p_--p_i}v^{\beta+p_i}dx\bigg).\end{gathered}$$

The finiteness of the integrals in is ensured by the fact that $v\in L^\infty{\left(}{\mathbb{R}}^n{\right)}$, $\eta v$ has compact support, and $\eta^{{\left(}\beta+p_-{\right)}/p_+}\in C^1{\left(}{\mathbb{R}}^n{\right)}$.

For any $\varepsilon>0$, we define $v_\varepsilon:=v+\varepsilon\overline\eta$, where $\overline\eta$ is a cutoff function on a neighborhood of the support of $\eta v$ such that $\overline\eta^{\beta+1}\in C^1{\left(}{\mathbb{R}}^n{\right)}$. Since $v\in D^{1,\operatorname{\overrightarrow{p}}}{\left(}{\mathbb{R}}^n{\right)}\cap L^\infty{\left(}{\mathbb{R}}^n{\right)}$, $\eta^{{\left(}\beta+p_-{\right)}/p_+}\in C^1{\left(}{\mathbb{R}}^n{\right)}$, and ${\left(}\beta+p_-{\right)}/p_+\le{\left(}\beta+p_-{\right)}/p_-=1+\beta/p_-$, we get ${\left(}\eta v_\varepsilon{\right)}^{\min{\left(}1,1+\beta/p_-{\right)}}\in D^{1,\operatorname{\overrightarrow{p}}}{\left(}{\mathbb{R}}^n{\right)}$. By a generalized version of the anisotropic Sobolev inequality (see Cîrstea–Vétois [@CirVet]\*[Lemma A.1]{}), we then obtain $$\label{Lem5Eq3}
\left\|\eta v_\varepsilon\right\|_{L^{\frac{n{\left(}\beta+p{\right)}}{n-p}}{\left(}{\mathbb{R}}^n{\right)}}^{\beta+p}\le C{\left(}\beta+p{\right)}^p\prod_{i=1}^n\left\|{\left(}\eta v_\varepsilon{\right)}^{\frac{\beta}{p_i}}\partial_{x_i}{\left(}\eta v_\varepsilon{\right)}\right\|_{L^{p_i}{\left(}{\mathbb{R}}^n{\right)}}^{\frac{p}{n}}<\infty$$ for some constant $C=C{\left(}n,\operatorname{\overrightarrow{p}}{\right)}$. For any $i=1,\dotsc,n$, we have $$\label{Lem5Eq4}
\int_{{\mathbb{R}}^n}{\left(}\eta v_\varepsilon{\right)}^\beta\left|\partial_{x_i}{\left(}\eta v_\varepsilon{\right)}\right|^{p_i}dx\le2^{p_i-1}{\left(}\int_{{\mathbb{R}}^n}\left|\partial_{x_i}\eta\right|^{p_i}\eta^\beta v_\varepsilon^{\beta+p_i}dx+\int_{{\mathbb{R}}^n}\left|\partial_{x_i} v_\varepsilon\right|^{p_i}\eta^{\beta+p_i}v_\varepsilon^\beta dx{\right)}.$$ Since $v\in D^{1,\operatorname{\overrightarrow{p}}}{\left(}{\mathbb{R}}^n{\right)}\cap L^\infty{\left(}{\mathbb{R}}^n{\right)}$, $\overline\eta^{\beta+1}\in C^1{\left(}{\mathbb{R}}^n{\right)}$, $\eta^{{\left(}\beta+p_-{\right)}/p_+}\in C^1{\left(}{\mathbb{R}}^n{\right)}$, and ${\left(}\beta+p_-{\right)}/p_+\le\beta+p_-$, we get $\eta^{\beta+p_i}v_\varepsilon^{\beta+1}\in D^{1,\operatorname{\overrightarrow{p}}}{\left(}{\mathbb{R}}^n{\right)}$. For any $i=1,\dotsc,n$, since $v_\varepsilon\equiv v+\varepsilon$ on the support of $\eta v$, testing with $\eta^{\beta+p_i}v_\varepsilon^{\beta+1}$ gives $$\begin{gathered}
\label{Lem5Eq5}
{\left(}\beta+1{\right)}\sum_{j=1}^n\int_{{\mathbb{R}}^n}\left|\partial_{x_j}v\right|^{p_j}\eta^{\beta+p_i}v_\varepsilon^\beta dx\le\Lambda\int_{{\mathbb{R}}^n}\eta^{\beta+p_i}v^{p^*-1}v_\varepsilon^{\beta+1}dx\\
-{\left(}\beta+p_i{\right)}\sum_{j=1}^n\int_{{\mathbb{R}}^n}\left|\partial_{x_j}v\right|^{p_j-2}{\left(}\partial_{x_j}v{\right)}{\left(}\partial_{x_j}\eta{\right)}\eta^{\beta+p_i-1}v_\varepsilon^{\beta+1}dx\,.\end{gathered}$$ For any $i,j=1,\dotsc,n$, Youngs inequality yields $$\begin{gathered}
\label{Lem5Eq6}
-{\left(}\beta+p_i{\right)}\left|\partial_{x_j}v\right|^{p_j-2}{\left(}\partial_{x_j}v{\right)}{\left(}\partial_{x_j}\eta{\right)}\eta^{\beta+p_i-1}v_\varepsilon^{\beta+1}\\
\le\frac{p_j-1}{p_j}\cdot{\left(}\beta+1{\right)}\left|\partial_{x_j}v\right|^{p_j}\eta^{\beta+p_i}v_\varepsilon^{\beta}+\frac{1}{p_j}\cdot\frac{{\left(}\beta+p_i{\right)}^{p_j}}{{\left(}\beta+1{\right)}^{p_j-1}}\left|\partial_{x_j}\eta\right|^{p_j}\eta^{\beta+p_i-p_j}v_\varepsilon^{\beta+p_j}.\end{gathered}$$ It follows from and that $$\begin{gathered}
\label{Lem5Eq7}
\sum_{j=1}^n\frac{1}{p_j}\int_{{\mathbb{R}}^n}\left|\partial_{x_j}v\right|^{p_j}\eta^{\beta+p_i}v_\varepsilon^\beta dx\le\Lambda{\left(}\beta+1{\right)}^{-1}\int_{{\mathbb{R}}^n}\eta^{\beta+p_i}v^{p^*-1}v_\varepsilon^{\beta+1}dx\\
+\sum_{j=1}^n\frac{1}{p_j}\cdot{\left(}\frac{\beta+p_i}{\beta+1}{\right)}^{p_j}\int_{{\mathbb{R}}^n}\left|\partial_{x_j}\eta\right|^{p_j}\eta^{\beta+p_i-p_j}v_\varepsilon^{\beta+p_j}dx\,.\end{gathered}$$ In particular, by and , we obtain $$\begin{gathered}
\label{Lem5Eq8}
\int_{{\mathbb{R}}^n}{\left(}\eta v_\varepsilon{\right)}^\beta\left|\partial_{x_i}{\left(}\eta v_\varepsilon{\right)}\right|^{p_i}dx\le C\bigg(
\Lambda{\left(}\beta+1{\right)}^{-1}\int_{{\mathbb{R}}^n}\eta^{\beta+p_i}v^{p^*-1}v_\varepsilon^{\beta+1}dx\\
+\sum_{j=1}^n\min{\left(}1,\beta+1{\right)}^{-p_j}\int_{{\mathbb{R}}^n}\left|\partial_{x_j}\eta\right|^{p_j}\eta^{\beta+p_i-p_j}v_\varepsilon^{\beta+p_j}dx+\varepsilon^{\beta+p_i}\int_{{\mathbb{R}}^n}\left|\partial_{x_i}\overline\eta\right|^{p_i}\eta^{\beta+p_i}\overline\eta^{\beta}dx\bigg)\end{gathered}$$ for some constant $C=C{\left(}n,\operatorname{\overrightarrow{p}}{\right)}$. Finally, since $\eta^{p_i}\le\eta^{p_-}$, we get by plugging into and passing to the limit as $\varepsilon\to0$. This ends the proof of Lemma \[Lem5\].

Now, we can prove Lemma \[Lem4\] by using Lemma \[Lem5\].

We denote $\beta:=\frac{n-p}{n}\gamma-p$. In particular, $\gamma>p_*-1$ is equivalent to $\beta>-1$. In connexion with the sets $\Omega_{{\overrightarrow{q}}}{\left(}\mathcal{I}_1,R_1,\mathcal{I}_2,R_2,\lambda{\right)}$, we define test functions of the form $$\label{Lem4Eq2}
\eta{\left(}x{\right)}:=\bigg[\overline\eta_{\lambda,\lambda'}\bigg(R_1^{-1}\sum_{i\in\mathcal{I}_1}\left|x_i\right|^{q_i}\bigg)\widetilde\eta_{\lambda,\lambda'}\bigg(R_2^{-1}\sum_{i\in\mathcal{I}_2}\left|x_i\right|^{q_i}\bigg)\bigg]^{\max{\left(}1,\frac{p_+}{\beta+p_-}{\right)}}$$ for all $x\in{\mathbb{R}}^n$, where $\overline\eta_{\lambda,\lambda'},\,\widetilde\eta_{\lambda,\lambda'}\in C^1{\left(}0,\infty{\right)}$ satisfy $0\le\overline\eta_{\lambda,\lambda'},\,\widetilde\eta_{\lambda,\lambda'}\le1$ in ${\left(}0,\infty{\right)}$, $\overline\eta_{\lambda,\lambda'}=1$ in ${\left[}0,1+\lambda{\right]}$, $\overline\eta_{\lambda,\lambda'}=0$ in ${\left[}1+\lambda',\infty{\right)}$, $\left|\overline\eta'_{\lambda,\lambda'}\right|\le2$ in ${\left[}1+\lambda,1+\lambda'{\right]}$, $\widetilde\eta_{\lambda,\lambda'}=1$ in ${\left[}1-\lambda,1+\lambda{\right]}$, $\widetilde\eta_{\lambda,\lambda'}=0$ in ${\left[}0,1-\lambda'{\right]}\cup{\left[}1+\lambda',\infty{\right)}$, and $\left|\widetilde\eta'_{\lambda,\lambda'}\right|\le2/{\left(}\lambda'-\lambda{\right)}$ in ${\left[}1-\lambda',1-\lambda{\right]}\cup{\left[}1+\lambda,1+\lambda'{\right]}$. With these properties of $\overline\eta_{\lambda,\lambda'}$ and $\widetilde\eta_{\lambda,\lambda'}$, we obtain $$0\le\eta\le1\text{ in }{\mathbb{R}}^n,\,\eta=1\text{ in }\Omega_{{\overrightarrow{q}}}{\left(}\mathcal{I}_1,R_1,\mathcal{I}_2,R_2,\lambda{\right)},\text{ and }\eta=0\text{ in }{\mathbb{R}}^n\backslash \Omega_{{\overrightarrow{q}}}{\left(}\mathcal{I}_1,R_1,\mathcal{I}_2,R_2,\lambda'{\right)}.$$ Since $\Omega_{{\overrightarrow{q}}}{\left(}\mathcal{I}_1,R_1,\mathcal{I}_2,R_2,\lambda'{\right)}\cap\operatorname{supp}{\left(}u{\right)}$ is bounded by assumption, we get that $\eta u$ has compact support. Moreover, since $q_i>1$ for all $i\in\mathcal{I}_1\cup\mathcal{I}_2$, we get $\eta^{{\left(}\beta+p_-{\right)}/p_+}\in C^1{\left(}{\mathbb{R}}^n{\right)}$ and $$\label{Lem4Eq3}
\left|\partial_{x_i}\eta{\left(}x{\right)}\right|^{p_i}\eta{\left(}x{\right)}^{\beta+p_--p_i}\le{\left(}\frac{4q_i}{\lambda'-\lambda}\max{\left(}1,\frac{p_+}{\beta+p_-}{\right)}{\right)}^{p_i}R_{\delta_i}^{-\frac{p_i}{q_i}}$$ for all $x\in\operatorname{supp}{\left(}\eta{\right)}$, where $\delta_i:=1$ if $i\in\mathcal{I}_1$, $\delta_i:=2$ if $i\in\mathcal{I}_2$. By applying Lemma \[Lem5\] with $v=\left|u\right|$ and $\eta$ as in , and using , we obtain $$\begin{gathered}
\label{Lem4Eq4}
{\left(}\int_{\Omega_{{\overrightarrow{q}}}{\left(}\mathcal{I}_1,R_1,\mathcal{I}_2,R_2,\lambda{\right)}}u^\gamma dx{\right)}^{\frac{n-p}{n}}\le C{\left(}\left|\beta\right|^p+1{\right)}\bigg(\Lambda{\left(}\beta+1{\right)}^{-1}\int_{\Omega_{{\overrightarrow{q}}}{\left(}\mathcal{I}_1,R_1,\mathcal{I}_2,R_2,\lambda'{\right)}}u^{\beta+p^*}dx\\
+\sum_{i\in\mathcal{I}_1\cup\mathcal{I}_2}\min{\left(}1,\beta+1{\right)}^{-p_i}{\left(}\lambda'-\lambda{\right)}^{-p_i}q_i^{p_i}R_{\delta_i}^{-\frac{p_i}{q_i}}\int_{\Omega_{{\overrightarrow{q}}}{\left(}\mathcal{I}_1,R_1,\mathcal{I}_2,R_2,\lambda'{\right)}}u^{\beta+p_i}dx\bigg)\end{gathered}$$ for some constant $C=C{\left(}n,\operatorname{\overrightarrow{p}}{\right)}$.

Now, we estimate the first integral in the right-hand side of . We claim that there exists a constant $C'=C{\left(}n,\operatorname{\overrightarrow{p}},K_0{\right)}$ such that $$\label{Lem4Eq5}
u{\left(}x{\right)}^{p^*-p_{i_0}}\le C'R_2^{-\frac{p_{i_0}}{q_{i_0}}}\quad\text{for all }x\in\Omega_{{\overrightarrow{q}}}{\left(}\mathcal{I}_1,R_1,\mathcal{I}_2,R_2,\lambda'{\right)},$$ where $K_0$ is the constant given by Lemma \[Lem3\] and $i_0\in\mathcal{I}_2$ is such that $$\label{Lem4Eq6}
\frac{q_{i_0}{\left(}p^*-p_{i_0}{\right)}}{p_{i_0}}=\max_{i\in\mathcal{I}_2}{\left(}\frac{q_i{\left(}p^*-p_i{\right)}}{p_i}{\right)}\,.$$ We prove this claim. For any $x\in\Omega_{{\overrightarrow{q}}}{\left(}\mathcal{I}_1,R_1,\mathcal{I}_2,R_2,\lambda'{\right)}$, since $\lambda'\le1/2$, we obtain $$\label{Lem4Eq7}
\frac{R_2}{2}\le\sum_{i\in\mathcal{I}_2}\left|x_i\right|^{q_i}\le n\cdot|x_{i{\left(}x{\right)}}|^{q_{i{\left(}x{\right)}}}\le n\cdot d_{\operatorname{\overrightarrow{p}}}{\left(}x,0{\right)}^{\frac{q_{i{\left(}x{\right)}}(p^*-p_{i{\left(}x{\right)}})}{\delta p_{i{\left(}x{\right)}}}},$$ where $i{\left(}x{\right)}\in\mathcal{I}_2$ is such that $|x_{i{\left(}x{\right)}}|^{q_{i{\left(}x{\right)}}}=\max{\left(}\left\{\left|x_i\right|^{q_i}:\,i\in\mathcal{I}_2\right\}{\right)}$, and the distance function $d_{\operatorname{\overrightarrow{p}}}$ and the real number $\delta$ are as in . Since $\Omega_{{\overrightarrow{q}}}{\left(}\mathcal{I}_1,R_1,\mathcal{I}_2,R_2,\lambda'{\right)}\cap B_{\operatorname{\overrightarrow{p}}}{\left(}0,\max{\left(}r,1{\right)}{\right)}=\emptyset$ by assumption, follows from , , and Lemma \[Lem3\]. In particular, implies $$\label{Lem4Eq9}
\int_{\Omega_{{\overrightarrow{q}}}{\left(}\mathcal{I}_1,R_1,\mathcal{I}_2,R_2,\lambda'{\right)}}u^{\beta+p^*}dx\le C'R_2^{-\frac{p_{i_0}}{q_{i_0}}}\int_{\Omega_{{\overrightarrow{q}}}{\left(}\mathcal{I}_1,R_1,\mathcal{I}_2,R_2,\lambda'{\right)}}u^{\beta+p_{i_0}}dx\,.$$ Finally, follows from , , and the fact that $\beta+1=\frac{n-p}{n}{\left(}\gamma-p_*+1{\right)}$ and $\beta+p_i=\gamma_i$. This ends the proof of Lemma \[Lem4\].

The iteration scheme {#Sec5}
====================

In this section, we describe the iteration scheme which leads to the proofs of our main results.

Let $\mathcal{I}_1$ and $\mathcal{I}_2$ be two disjoint subsets of $\left\{1,\dotsc,n\right\}$, $\mathcal{I}_2\ne\emptyset$, and ${\overrightarrow{q}}={\left(}q_i{\right)}_{i\in\mathcal{I}_1\cup\mathcal{I}_2}$ be such that $q_i>1$ for all $i\in\mathcal{I}_1\cup\mathcal{I}_2$. The idea is to apply Lemma \[Lem4\] by induction. For any $\gamma>p_*-1$, Lemma \[Lem4\] provides an estimate of the $L^\gamma$–norm of $u$ with respect to the set of $L^{\gamma_{i_1}}$–norms of $u$, where $\gamma_{i_1}:=\frac{n-p}{n}\gamma+p_{i_1}-p$ for all $i_1\in\mathcal{I}_1\cup\mathcal{I}_2$. If $\gamma_{i_1}>p_*-1$, then another application of Lemma \[Lem4\] gives estimates of the $L^{\gamma_{i_1}}$–norms of $u$ with respect to the set of $L^{\gamma_{i_1i_2}}$–norms of $u$, where $\gamma_{i_1i_2}:=\frac{n-p}{n}\gamma_{i_1}+p_{i_2}-p$, etc... By induction, we define $$\label{Eq11}
\gamma_{i_1,\dotsc,i_{j+1}}:=\frac{n-p}{n}\gamma_{i_1,\dotsc,i_j}+p_{i_{j+1}}-p$$ for all $j\in{\mathbb{N}}$ and $i_1,\dotsc,i_{j+1}\in\mathcal{I}_1\cup\mathcal{I}_2$, with the convention that $\gamma_{i_1,\dotsc,i_j}:=\gamma$ if $j=0$. In particular, we obtain the formula $$\label{Eq12}
\gamma_{i_1,\dotsc,i_k}={\left(}\frac{n-p}{n}{\right)}^k\gamma+\sum_{j=1}^k{\left(}\frac{n-p}{n}{\right)}^{k-j}{\left(}p_{i_j}-p{\right)}$$ for all $k\in{\mathbb{N}}$. The stopping condition in our induction argument is $\gamma_{i_1,\dotsc,i_k}<\frac{n}{p}{\left(}p_\varepsilon-p{\right)}$, where $$\label{Eq13}
p_\varepsilon:={\left(}1+\varepsilon{\right)}p_0\,,\quad p_0:=\max{\left(}p_*,\left\{p_i\,:\,i\in\mathcal{I}_1\cup\mathcal{I}_2\right\}{\right)},$$ and $\varepsilon$ is a fixed real number in ${\left(}0,1{\right)}$. Note that $\frac{n}{p}{\left(}p_\varepsilon-p{\right)}>\frac{n}{p}{\left(}p_*-p{\right)}=p_*-1$ so that we can apply Lemma \[Lem4\] as long as our stopping condition is not satisfied. For any $k\ge1$, we let $\Phi_{k,\gamma,\varepsilon}$ be the set of all sequences of indices for which our induction argument stops after exactly $k$ iterations, namely $$\begin{gathered}
\label{Eq14}
\Phi_{k,\gamma,\varepsilon}:=\Big\{{\left(}i_1,\dotsc,i_k{\right)}\in{\left(}\mathcal{I}_1\cup\mathcal{I}_2{\right)}^k\,:\quad\gamma_{i_1,\dotsc,i_j}\ge\frac{n}{p}{\left(}p_\varepsilon-p{\right)}\text{ for all }j=0,\dotsc,k-1\\
\text{and}\quad\gamma_{i_1,\dotsc,i_k}<\frac{n}{p}{\left(}p_\varepsilon-p{\right)}\Big\}.\end{gathered}$$ The following result provides a control on the number of iterations in our induction argument.

\[Lem6\] Let $\mathcal{I}_1$ and $\mathcal{I}_2$ be two disjoint subsets of $\left\{1,\dotsc,n\right\}$, $\mathcal{I}_2\ne\emptyset$, and ${\overrightarrow{q}}={\left(}q_i{\right)}_{i\in\mathcal{I}_1\cup\mathcal{I}_2}$ be such that $q_i>1$ for all $i\in\mathcal{I}_1\cup\mathcal{I}_2$. Then for any $\varepsilon>0$, $\gamma\ge\frac{n}{p}{\left(}p_\varepsilon-p{\right)}$, $k\in{\mathbb{N}}$, and ${\left(}i_1,\dotsc,i_k{\right)}\in{\left(}\mathcal{I}_1\cup\mathcal{I}_2{\right)}^k$, we have $$\label{Lem6Eq1}
\gamma_{i_1,\dotsc,i_k}>\frac{n}{p}{\left(}p_\varepsilon-p{\right)}\quad\text{if }k<k_{\gamma,\varepsilon}^-\quad\text{and}\quad\gamma_{i_1,\dotsc,i_k}<\frac{n}{p}{\left(}p_\varepsilon-p{\right)}\quad\text{if }k\ge k_{\gamma,\varepsilon}^+\,,$$ where $\gamma_{i_1,\dotsc,i_k}$ is as in , $p_\varepsilon$ is as in , and $k_{\gamma,\varepsilon}^-$ and $k_{\gamma,\varepsilon}^+$ are the smallest and largest natural numbers, respectively, such that $$\label{Lem6Eq2}
\frac{n}{p}{\left(}\frac{n}{n-p}{\right)}^{k_{\gamma,\varepsilon}^+-1}\varepsilon p_0<\gamma<\frac{n}{p}{\left(}\frac{n}{n-p}{\right)}^{k_{\gamma,\varepsilon}^-}{\left(}p_\varepsilon-p_-{\right)},$$ where $p_-:=\min{\left(}\left\{p_i\in\operatorname{\overrightarrow{p}}\right\}{\right)}$. In particular, we have $\Phi_{k,\gamma,\varepsilon}=\emptyset$ for all $k<k_{\gamma,\varepsilon}^-$ and $k>k_{\gamma,\varepsilon}^+$, where $\Phi_{k,\gamma,\varepsilon}$ is as in .

Since $p_-\le p_{i_j}\le p_0$ for all $j=1,\dotsc,k$, it follows from that $$\label{Lem6Eq3}
-\sum_{j=1}^k{\left(}\frac{n-p}{n}{\right)}^{k-j}{\left(}p-p_-{\right)}\le\gamma_{i_1,\dotsc,i_k}-{\left(}\frac{n-p}{n}{\right)}^k\gamma\le\sum_{j=1}^k{\left(}\frac{n-p}{n}{\right)}^{k-j}{\left(}p_0-p{\right)}.$$ Moreover, by a simple calculation, we obtain $$\label{Lem6Eq4}
\sum_{j=1}^k{\left(}\frac{n-p}{n}{\right)}^{k-j}=\frac{n}{p}{\left(}1-{\left(}\frac{n-p}{n}{\right)}^k{\right)}<\frac{n}{p}\,.$$ It follows from and that $$\label{Lem6Eq5}
-\frac{n}{p}{\left(}p-p_-{\right)}<\gamma_{i_1,\dotsc,i_k}-{\left(}\frac{n-p}{n}{\right)}^k\gamma<\frac{n}{p}{\left(}p_0-p{\right)}.$$ Finally, follows from together with the definitions of $k_{\gamma,\varepsilon}^-$ and $k_{\gamma,\varepsilon}^+$.

Now, we can prove the main result of this section.

\[Lem7\] Assume that $p_+<p^*$. Let $f:{\mathbb{R}}^n\times{\mathbb{R}}\to{\mathbb{R}}$ be a Caratheodory function such that holds true, $u$ be a solution of , and $\kappa$, $r$, and $K_0$ be as in Lemma \[Lem3\]. Let $\mathcal{I}_1$ and $\mathcal{I}_2$ be two disjoint subsets of $\left\{1,\dotsc,n\right\}$, $\mathcal{I}_2\ne\emptyset$, and ${\overrightarrow{q}}={\left(}q_i{\right)}_{i\in\mathcal{I}_1\cup\mathcal{I}_2}$ be such that $q_i>1$ for all $i\in\mathcal{I}_1\cup\mathcal{I}_2$. Then there exists a constant $c_1=c_1{\left(}n,\operatorname{\overrightarrow{p}},\Lambda,K_0{\right)}>1$ such that for any $\varepsilon\in{\left(}0,1{\right)}$, $\gamma>\frac{n}{p}{\left(}p_\varepsilon-p{\right)}$, and $R_1,R_2>0$ such that $\Omega_{{\overrightarrow{q}}}{\left(}\mathcal{I}_1,R_1,\mathcal{I}_2,R_2,1/2{\right)}\cap B_{\operatorname{\overrightarrow{p}}}{\left(}0,\max{\left(}r,1{\right)}{\right)}=\emptyset$ and $\Omega_{{\overrightarrow{q}}}{\left(}\mathcal{I}_1,R_1,\mathcal{I}_2,R_2,1/2{\right)}\cap\operatorname{supp}{\left(}u{\right)}$ is bounded, we have $$\begin{gathered}
\label{Lem7Eq1}
\left\|u\right\|_{L^{\gamma}{\left(}\Omega_{{\overrightarrow{q}}}{\left(}\mathcal{I}_1,R_1,\mathcal{I}_2,R_2,\lambda_{0,\gamma,\varepsilon}{\right)}{\right)}}\le c_1^{\frac{1}{\varepsilon}}\max{\left(}q_i{\right)}^{\frac{1}{\varepsilon}\cdot\frac{n}{n-p}}{\left(}p_\varepsilon-p_*{\right)}^{-\frac{1}{\varepsilon}\cdot\frac{n}{n-p}}\\
\times\max_{{\left(}i_1,\dotsc,i_k{\right)}\in\Phi_{\gamma,\varepsilon}}{\left(}{\left(}\prod_{j=1}^kR_{\delta_{i_j}}^{-\frac{1}{\gamma}{\left(}\frac{n}{n-p}{\right)}^j\frac{p_{i_j}}{q_{i_j}}}{\right)}\left\|u\right\|_{L^{\gamma_{i_1,\dotsc,i_k}}{\left(}\Omega_{{\overrightarrow{q}}}{\left(}\mathcal{I}_1,R_1,\mathcal{I}_2,R_2,\lambda_{k,\gamma,\varepsilon}{\right)}{\right)}}^{\frac{\gamma_{i_1,\dotsc,i_k}}{\gamma}{\left(}\frac{n}{n-p}{\right)}^k}{\right)},\end{gathered}$$ where $\delta_{i_j}:=1$ if $i_j\in\mathcal{I}_1$, $\delta_{i_j}:=2$ if $i_j\in\mathcal{I}_2$, $\gamma_{i_1,\dotsc,i_k}$ is as in , $p_\varepsilon$ is as in , $\Omega_{{\overrightarrow{q}}}{\left(}\mathcal{I}_1,R_1,\mathcal{I}_2,R_2,\lambda_k{\right)}$ is as in , and $$\label{Lem7Eq2}
\lambda_{k,\gamma,\varepsilon}:=\frac{1}{4}\big(1+2^{k-k_{\gamma,\varepsilon}^+-1}\big)\quad\text{and}\quad\Phi_{\gamma,\varepsilon}:=\bigcup_{k=k_{\gamma,\varepsilon}^-}^{k_{\gamma,\varepsilon}^+}\Phi_{k,\gamma,\varepsilon}$$ with $k_{\gamma,\varepsilon}^-$ and $k_{\gamma,\varepsilon}^+$ as in Lemma \[Lem6\], and $\Phi_{k,\gamma,\varepsilon}$ as in .

Applying Lemma \[Lem4\] by induction with the stopping condition $\gamma_{i_1,\dotsc,i_k}<\frac{n}{p}{\left(}p_\varepsilon-p{\right)}$ gives $$\begin{gathered}
\label{Lem7Eq3}
\left\|u\right\|_{L^{\gamma}{\left(}\Omega_{{\overrightarrow{q}}}{\left(}\mathcal{I}_1,R_1,\mathcal{I}_2,R_2,\lambda_{0,\gamma,\varepsilon}{\right)}{\right)}}\le\max_{{\left(}i_1,\dotsc,i_k{\right)}\in\Phi_{\gamma,\varepsilon}}\Big(\mathcal{A}_{k,\gamma}\times\mathcal{B}_{i_1,\dotsc,i_{k-1},\gamma}\times\mathcal{C}_{i_1,\dotsc,i_k,\gamma}\times\mathcal{D}_{i_1,\dotsc,i_k,\gamma,\varepsilon}\\
\times{\left(}\prod_{j=1}^kR_{\delta_{i_j}}^{-\frac{1}{\gamma}{\left(}\frac{n}{n-p}{\right)}^j\frac{p_{i_j}}{q_{i_j}}}{\right)}\left\|u\right\|_{L^{\gamma_{i_1,\dotsc,i_k}}{\left(}\Omega_{{\overrightarrow{q}}}{\left(}\mathcal{I}_1,R_1,\mathcal{I}_2,R_2,\lambda_{k,\gamma,\varepsilon}{\right)}{\right)}}^{\frac{\gamma_{i_1,\dotsc,i_k}}{\gamma}{\left(}\frac{n}{n-p}{\right)}^k}\Big),\end{gathered}$$ where $\lambda_{k,\gamma,\varepsilon}$ and $\Phi_{\gamma,\varepsilon}$ are as in , and $$\begin{aligned}
&\mathcal{A}_{k,\gamma}:=\big(c_0\cdot\max\big(q_i^{\frac{np_i}{n-p}}\big)\big)^{\frac{1}{\gamma}\overset{k-1}{\underset{j=0}{\sum}}{\left(}\frac{n}{n-p}{\right)}^j},\quad\mathcal{B}_{i_1,\dotsc,i_{k-1},\gamma}:=\prod_{j=1}^k\gamma_{i_1,\dotsc,i_{j-1}}^{\frac{p}{\gamma}{\left(}\frac{n}{n-p}{\right)}^j},\allowdisplaybreaks\\
&\mathcal{C}_{i_1,\dotsc,i_{k-1},\gamma}:=\prod_{j=1}^k\min{\left(}1,\gamma_{i_1,\dotsc,i_{j-1}}-p_*+1{\right)}^{-\frac{p_{i_j}}{\gamma}{\left(}\frac{n}{n-p}{\right)}^j},\allowdisplaybreaks\\
&\mathcal{D}_{i_1,\dotsc,i_k,\gamma,\varepsilon}:=\prod_{j=1}^k{\left(}\lambda_{j,\gamma,\varepsilon}-\lambda_{j-1,\gamma,\varepsilon}{\right)}^{-\frac{p_{i_j}}{\gamma}{\left(}\frac{n}{n-p}{\right)}^j}.\end{aligned}$$ Now, we fix ${\left(}i_1,\dotsc,i_k{\right)}\in\Phi_{\gamma,\varepsilon}$ and estimate each of the terms in the right-hand side of .

By using the fact that $k\le k_{\gamma,\varepsilon}^+$ and applying , we obtain $$\label{Lem7Eq4}
\sum_{j=0}^{k-1}{\left(}\frac{n}{n-p}{\right)}^j=\frac{n-p}{p}{\left[}{\left(}\frac{n}{n-p}{\right)}^k-1{\right]}<\frac{n}{p}{\left(}\frac{n}{n-p}{\right)}^{k_{\gamma,\varepsilon}^+-1}<\frac{\gamma}{\varepsilon p_0}\,.$$ Since $c_0>1$, $q_i>1$, and $p_i\le p_0$ for all $i\in\mathcal{I}_1\cup\mathcal{I}_2$, it follows from that $$\label{Lem7Eq5}
\mathcal{A}_{k,\gamma}<c_0^{\frac{1}{\varepsilon p_0}}\max{\left(}q_i{\right)}^{\frac{1}{\varepsilon}\cdot\frac{n}{n-p}}.$$

For any $j=1,\dotsc,k$, since $p_i\le p_0$ for all $i\in\mathcal{I}_1\cup\mathcal{I}_2$, by , , and , we obtain $$\label{Lem7Eq6}
\gamma_{i_1,\dotsc,i_{j-1}}\le{\left(}\frac{n-p}{n}{\right)}^{j-1}\gamma+\frac{n}{p}{\left(}p_0-p{\right)}\le C\max\bigg(1,{\left(}\frac{n}{n-p}{\right)}^{k_{\gamma,\varepsilon}^--j}\bigg)$$ for some constant $C=C{\left(}n,\operatorname{\overrightarrow{p}}{\right)}>1$. It follows from that $$\label{Lem7Eq7}
\mathcal{B}_{i_1,\dotsc,i_{k-1},\gamma}\le C^{\frac{p}{\gamma}\overset{k}{\underset{j=1}{\sum}}{\left(}\frac{n}{n-p}{\right)}^j}{\left(}\frac{n}{n-p}{\right)}^{\frac{p}{\gamma}\overset{k_{\gamma,\varepsilon}^-}{\underset{j=1}{\sum}}{\left(}\frac{n}{n-p}{\right)}^j{\left(}k_{\gamma,\varepsilon}^--j{\right)}}.$$ A simple calculation gives $$\overset{k_{\gamma,\varepsilon}^-}{\underset{j=1}{\sum}}{\left(}\frac{n}{n-p}{\right)}^j{\left(}k_{\gamma,\varepsilon}^--j{\right)}=\frac{n^2}{p^2}{\left[}{\left(}\frac{n}{n-p}{\right)}^{k_{\gamma,\varepsilon}^--1}-\frac{p}{n}{\left(}k_{\gamma,\varepsilon}^--1{\right)}-1{\right]}<\frac{n^2}{p^2}{\left(}\frac{n}{n-p}{\right)}^{k_{\gamma,\varepsilon}^--1},$$ and hence by definition of $k_{\gamma,\varepsilon}^-$, we obtain $$\label{Lem7Eq8}
\overset{k_{\gamma,\varepsilon}^-}{\underset{j=1}{\sum}}{\left(}\frac{n}{n-p}{\right)}^j{\left(}k_{\gamma,\varepsilon}^--j{\right)}\le\frac{n}{p}\cdot\frac{\gamma}{p_\varepsilon-p_-}<\frac{n}{p}\cdot\frac{\gamma}{p_*-p_-}\,.$$ It follows from , , and that $$\label{Lem7Eq9}
\mathcal{B}_{i_1,\dotsc,i_{k-1},\gamma}\le C^{\frac{p^*}{\varepsilon p_0}}{\left(}\frac{n}{n-p}{\right)}^{\frac{n}{p_*-p_-}}.$$

Since $p_*-1=\frac{n}{p}{\left(}p_*-p{\right)}$ and $\gamma_{i_1,\dotsc,i_{j-1}}>\frac{n}{p}{\left(}p_\varepsilon-p{\right)}$ for all $j=1,\dotsc,k$, we obtain $$\label{Lem7Eq10}
\mathcal{C}_{i_1,\dotsc,i_{k-1},\gamma}\le\min\Big(1,\frac{n}{p}{\left(}p_\varepsilon-p_*{\right)}\Big)^{-\frac{1}{\gamma}\overset{k}{\underset{j=1}{\sum}}p_{i_j}{\left(}\frac{n}{n-p}{\right)}^j}.$$ Since $p_{i_j}\le p_0$ for all $j=1,\dotsc,k$, it follows from and that $$\label{Lem7Eq11}
\mathcal{C}_{i_1,\dotsc,i_{k-1},\gamma}\le\min\Big(1,\frac{n}{p}{\left(}p_\varepsilon-p_*{\right)}\Big)^{-\frac{1}{\varepsilon}\cdot\frac{n}{n-p}}.$$

By and since $k\le k_{\gamma,\varepsilon}^+$ and $p_{i_j}\le p_0$ for all $j=1,\dotsc,k$, we obtain $$\label{Lem7Eq12}
\mathcal{D}_{i_1,\dotsc,i_k,\gamma,\varepsilon}\le2^{\frac{1}{\gamma}\overset{k}{\underset{j=1}{\sum}}p_{i_j}{\left(}\frac{n}{n-p}{\right)}^j{\left(}k_{\gamma,\varepsilon}^+-j+4{\right)}}\le2^{\frac{p_0}{\gamma}\overset{k_{\gamma,\varepsilon}^+}{\underset{j=1}{\sum}}{\left(}\frac{n}{n-p}{\right)}^j{\left(}k_{\gamma,\varepsilon}^+-j+4{\right)}}.$$ We find $$\begin{aligned}
\overset{k_{\gamma,\varepsilon}^+}{\underset{j=1}{\sum}}{\left(}\frac{n}{n-p}{\right)}^j{\left(}k_{\gamma,\varepsilon}^+-j+4{\right)}&=\frac{n}{p}{\left[}{\left(}\frac{n}{n-p}{\right)}^{k_{\gamma,\varepsilon}^+}{\left(}3+\frac{n}{p}{\right)}-k_{\gamma,\varepsilon}^+-3-\frac{n}{p}{\right]}\\
&<\frac{n}{p}{\left(}\frac{n}{n-p}{\right)}^{k_{\gamma,\varepsilon}^+}{\left(}3+\frac{n}{p}{\right)},\end{aligned}$$ and hence by , we obtain $$\label{Lem7Eq13}
\overset{k_{\gamma,\varepsilon}^+}{\underset{j=1}{\sum}}{\left(}\frac{n}{n-p}{\right)}^j{\left(}k_{\gamma,\varepsilon}^+-j+4{\right)}<\frac{\gamma}{\varepsilon p_0}\cdot\frac{n}{n-p}{\left(}3+\frac{n}{p}{\right)}.$$ It follows from and that $$\label{Lem7Eq14}
\mathcal{D}_{i_1,\dotsc,i_k,\gamma,\varepsilon}\le2^{\frac{1}{\varepsilon}\cdot\frac{n}{n-p}{\left(}3+\frac{n}{p}{\right)}}.$$

The estimate follows from , , , , and .

The vanishing result {#Sec6}
====================

In this section, we prove a vanishing result which will give Point (i) in Theorem \[Th3\]. We define $$\label{Eq15}
\overline{p}_0:=\max{\left(}p_*,\left\{p_i\in\operatorname{\overrightarrow{p}}\,:\,i\in\Theta\right\}{\right)},$$ where $p_*$ is as in and $\Theta$ is the set of all indices $i$ such that $$\label{Eq16}
\big(p_i-p_--\frac{n}{p}{\left(}p_i-p_*{\right)}\big)\sum_{j=1}^n\max{\left(}\frac{p_i-p_j}{p_j},0{\right)}\ge{\left(}p_*-1{\right)}{\left(}p_i-p_-{\right)}$$ with $p_-:=\min{\left(}\left\{p_i\in\operatorname{\overrightarrow{p}}\right\}{\right)}$. We define $\mathcal{I}_0$ as the set of all indices $i$ such that $p_i>\overline{p}_0$.

When $p_+>p_*$, one easily sees that the condition does not hold true for $p_i=p_+$, and hence we have $\overline{p}_0<p_+$ and $\mathcal{I}_0\ne\emptyset$.

We prove the following result.

\[Th6\] Assume that $p_*<p_+<p^*$. Let $f:{\mathbb{R}}^n\times{\mathbb{R}}\to{\mathbb{R}}$ be a Caratheodory function such that holds true and $u$ be a solution of . Then there exists a constant $R_0=R_0{\left(}n,\operatorname{\overrightarrow{p}},\Lambda,u{\right)}$ such that $u{\left(}x{\right)}=0$ for all $x\in{\mathbb{R}}^n$ such that $\sum_{i\in \mathcal{I}_0}\left|x_i\right|\ge R_0$.

The proof of Theorem \[Th6\] is based on the following result, which we obtain by applying the iteration scheme in Section \[Sec5\].

\[Lem8\] Assume that $p_*<p_+<p^*$. Let $f:{\mathbb{R}}^n\times{\mathbb{R}}\to{\mathbb{R}}$ be a Caratheodory function such that holds true, $u$ be a solution of , and $\kappa$, $r$, and $K_0$ be as in Lemma \[Lem3\]. Let $\overline{p}_0$ be as in and $p_0\in\operatorname{\overrightarrow{p}}$ be such that $$\label{Lem8Eq1}
p_0>\overline{p}_0\quad\text{and}\quad R_i{\left(}u{\right)}<\infty\text{ for all indices }i\text{ such that }p_i>p_0\,,$$ where $$\label{Lem8Eq2}
R_i{\left(}u{\right)}:=\sup{\left(}\left\{\left|x_i\right|\,:\,x\in\operatorname{supp}{\left(}u{\right)}\right\}{\right)}.$$ Let $\mathcal{I}_1$, $\mathcal{I}_2$ be the sets of indices $i$ such that $p_i<p_0$, $p_i=p_0$, respectively. For any $\varepsilon,\lambda\in{\left(}0,1{\right)}$ and $R>1$, we define $$\label{Lem8Eq3}
A_\varepsilon{\left(}R,\lambda{\right)}:=\Omega_{{\overrightarrow{q}}}{\left(}\mathcal{I}_1,R^{1/\varepsilon},\mathcal{I}_2,R,\lambda{\right)}\quad\text{with}\quad q_i:=\left\{\begin{aligned}&\frac{p_\varepsilon p_i}{p_\varepsilon-p_i}&&\text{if }i\in\mathcal{I}_1\,,\\&p_\varepsilon&&\text{if }i\in\mathcal{I}_2\,,\end{aligned}\right.$$ where $p_\varepsilon:={\left(}1+\varepsilon{\right)}p_0$. If $A_\varepsilon{\left(}R,1/2{\right)}\cap B_{\operatorname{\overrightarrow{p}}}{\left(}0,\max{\left(}r,1{\right)}{\right)}=\emptyset$, then $$\label{Lem8Eq4}
\left\|u\right\|_{L^\infty{\left(}A_\varepsilon{\left(}R,1/4{\right)}{\right)}}\le \big(c_2R^{-\frac{1}{p_\varepsilon}}\big)^{\frac{1}{\varepsilon}}$$ for some constant $c_2=c_2(n,\operatorname{\overrightarrow{p}},\Lambda,K_0,\left\|u\right\|_{L^{p_*-1,\infty}{\left(}{\mathbb{R}}^n{\right)}},R_0{\left(}u{\right)})$, where $$\label{Lem8Eq5}
R_0{\left(}u{\right)}:=\max{\left(}\left\{R_i{\left(}u{\right)}:i\in\left\{1,\dotsc,n\right\}\backslash{\left(}\mathcal{I}_1\cup\mathcal{I}_2{\right)}\right\}{\right)}.$$

As is easily seen, we have $1<q_i<s_0$ for some constant $s_0=s_0{\left(}\operatorname{\overrightarrow{p}}{\right)}$. Moreover, by , we obtain that $p_\varepsilon-p_*>p_0-p_*>0$ and $A_\varepsilon{\left(}R,1/2{\right)}\cap\operatorname{supp}{\left(}u{\right)}$ is bounded. By Lemma \[Lem7\], we then get that there exists a constant $\widetilde{c}_1=\widetilde{c}_1{\left(}n,\operatorname{\overrightarrow{p}},\Lambda,K_0{\right)}$ such that for any $\gamma>\frac{n}{p}{\left(}p_\varepsilon-p{\right)}$, we have $$\label{Lem8Eq6}
\left\|u\right\|_{L^{\gamma}{\left(}A_\varepsilon{\left(}R,\lambda_{0,\gamma,\varepsilon}{\right)}{\right)}}\le \widetilde{c}_1^{\,\frac{1}{\varepsilon}}\max_{{\left(}i_1,\dotsc,i_k{\right)}\in\Phi_{\gamma,\varepsilon}}\Big(R^{-\sigma_{i_1,\dotsc,i_k,\gamma,\varepsilon}}\left\|u\right\|_{L^{\gamma_{i_1,\dotsc,i_k}}{\left(}A_\varepsilon{\left(}R,\lambda_{k,\gamma,\varepsilon}{\right)}{\right)}}^{\frac{\gamma_{i_1,\dotsc,i_k}}{\gamma}{\left(}\frac{n}{n-p}{\right)}^k}\Big)$$ provided that $A_\varepsilon{\left(}R,1/2{\right)}\cap B_{\operatorname{\overrightarrow{p}}}{\left(}0,\max{\left(}r,1{\right)}{\right)}=\emptyset$, where $\gamma_{i_1,\dotsc,i_k}$ is as in , $\lambda_{k,\gamma,\varepsilon}$ and $\Phi_{\gamma,\varepsilon}$ are as in , and $$\label{Lem8Eq7}
\sigma_{i_1,\dotsc,i_k,\gamma,\varepsilon}:=\frac{1}{\varepsilon\gamma p_\varepsilon}\sum_{j=1}^k{\left(}\frac{n}{n-p}{\right)}^j{\left(}p_\varepsilon-p_{i_j}{\right)}\,.$$

We claim that for any $\nu\in{\left(}0,1{\right)}$ and ${\left(}i_1,\dotsc,i_k{\right)}\in\Phi_{\gamma,\varepsilon}$, there exists a constant $c_\nu=c\big(n,\operatorname{\overrightarrow{p}},K_0,\left\|u\right\|_{L^{p_*-1,\infty}{\left(}{\mathbb{R}}^n{\right)}},R_0{\left(}u{\right)},\nu\big)$ such that $$\label{Lem8Eq8}
\left\|u\right\|_{L^{\gamma_{i_1,\dotsc,i_k}}{\left(}A_\varepsilon{\left(}R,\lambda_{k,\gamma,\varepsilon}{\right)}{\right)}}^{\frac{\gamma_{i_1,\dotsc,i_k}}{\gamma}{\left(}\frac{n}{n-p}{\right)}^k}\le c_\nu^{\frac{1}{\varepsilon}}R^{\tau_{i_1,\dotsc,i_k,\gamma,\varepsilon,\nu}},$$ where $$\label{Lem8Eq9}
\tau_{i_1,\dotsc,i_k,\gamma,\varepsilon,\nu}:=\max\bigg(0\,,\frac{1}{\varepsilon\gamma p_\varepsilon}{\left(}\frac{n}{n-p}{\right)}^k{\left(}1-\frac{\gamma_{i_1,\dotsc,i_k}}{p_*-1+\nu}{\right)}\sum_{i\in\mathcal{I}_1\cup\mathcal{I}_2}\frac{p_\varepsilon-p_i}{p_i}\bigg).$$ We separate two cases:

-   Case 1: $p_*-1+\nu\le\gamma_{i_1,\dotsc,i_k}<\frac{n}{p}{\left(}p_\varepsilon-p{\right)}$ (in which case $\tau_{i_1,\dotsc,i_k,\gamma,\varepsilon,\nu}=0$).

-   Case 2: $\gamma_{i_1,\dotsc,i_k}<p_*-1+\nu$ (in which case $\tau_{i_1,\dotsc,i_k,\gamma,\varepsilon,\nu}>0$).

We begin with proving in Case 1. By interpolation (see, for instance, Grafakos [@Gra]\*[Proposition 1.1.14]{}), and by Lemmas \[Lem1\] and \[Lem3\], we obtain $$\begin{aligned}
\label{Lem8Eq10}
\left\|u\right\|_{L^{\gamma_{i_1,\dotsc,i_k}}{\left(}A_\varepsilon{\left(}R,\lambda_{k,\gamma,\varepsilon}{\right)}{\right)}}&\le{\left(}\frac{\gamma_{i_1,\dotsc,i_k}}{\gamma_{i_1,\dotsc,i_k}-p_*+1}{\right)}^{\frac{1}{\gamma_{i_1,\dotsc,i_k}}}\left\|u\right\|_{L^{p_*-1,\infty}{\left(}A_\varepsilon{\left(}R,\lambda_{k,\gamma,\varepsilon}{\right)}{\right)}}^{\frac{p_*-1}{\gamma_{i_1,\dotsc,i_k}}}\left\|u\right\|_{L^\infty{\left(}A_\varepsilon{\left(}R,\lambda_{k,\gamma,\varepsilon}{\right)}{\right)}}^{1-\frac{p_*-1}{\gamma_{i_1,\dotsc,i_k}}}\nonumber\\
&\le C\nu^{\frac{-1}{\gamma_{i_1,\dotsc,i_k}}}\le C\nu^{\frac{-1}{p_*-1}}\end{aligned}$$ for some constant $C=C\big(n,\operatorname{\overrightarrow{p}},K_0,\left\|u\right\|_{L^{p_*-1,\infty}{\left(}{\mathbb{R}}^n{\right)}}\big)$. Moreover, since $\gamma_{i_1,\dotsc,i_k}<\frac{n}{p}{\left(}p_\varepsilon-p{\right)}$ and $k\le k^+_{\gamma,\varepsilon}$, by , we get $$\label{Lem8Eq11}
\frac{\gamma_{i_1,\dotsc,i_k}}{\gamma}{\left(}\frac{n}{n-p}{\right)}^k\le\frac{1}{\varepsilon p_0}\cdot\frac{n}{n-p}{\left(}p_\varepsilon-p{\right)}.$$ Then follows from and .

Now, suppose that we are in Case 2. By and since $\gamma_{i_1,\dotsc,i_{k-1}}\ge\frac{n}{p}{\left(}p_\varepsilon-p{\right)}$ and $p_\varepsilon>p_*$, we obtain $$\label{Lem8Eq12}
\gamma_{i_1,\dotsc,i_k}\ge\frac{n}{p}{\left(}p_\varepsilon-p{\right)}+p_--p_\varepsilon>\frac{n}{p}{\left(}p_*-p{\right)}+p_--p_*=p_--1\,.$$ By Hölder’s inequality, we then get $$\label{Lem8Eq13}
\left\|u\right\|_{L^{\gamma_{i_1,\dotsc,i_k}}{\left(}A_\varepsilon{\left(}R,\lambda_{k,\gamma,\varepsilon}{\right)}{\right)}}\le\left|A_\varepsilon{\left(}R,\lambda_{k,\gamma,\varepsilon}{\right)}\cap\operatorname{supp}{\left(}u{\right)}\right|^{\frac{1}{\gamma_{i_1,\dotsc,i_k}}-\frac{1}{p_*-1+\nu}}\left\|u\right\|_{L^{p_*-1+\nu}{\left(}A_\varepsilon{\left(}R,\lambda_{k,\gamma,\varepsilon}{\right)}{\right)}}.$$ Direct computations yield $$\label{Lem8Eq14}
\left|A_\varepsilon{\left(}R,\lambda_{k,\gamma,\varepsilon}{\right)}\cap\operatorname{supp}{\left(}u{\right)}\right|\le CR^{\frac{1}{\varepsilon p_\varepsilon}\cdot\underset{i\in\mathcal{I}_1\cup\mathcal{I}_2}{\sum}\frac{p_\varepsilon-p_i}{p_i}}$$ for some constant $C=C\big(n,\operatorname{\overrightarrow{p}},R_0{\left(}u{\right)}\big)$, where $R_0{\left(}u{\right)}$ is as in . Similarly to and , we obtain $$\label{Lem8Eq15}
\left\|u\right\|_{L^{p_*-1+\nu}{\left(}A_\varepsilon{\left(}R,\lambda_{k,\gamma,\varepsilon}{\right)}{\right)}}\le C\nu^{\frac{-1}{p_*-1}}$$ for some constant $C=C\big(n,\operatorname{\overrightarrow{p}},K_0,\left\|u\right\|_{L^{p_*-1,\infty}{\left(}{\mathbb{R}}^n{\right)}}\big)$, and $$\label{Lem8Eq16}
\frac{\gamma_{i_1,\dotsc,i_k}}{\gamma}{\left(}\frac{n}{n-p}{\right)}^k<\frac{1}{\varepsilon p_0}\cdot\frac{p}{n-p}{\left(}p_*-1+\nu{\right)}.$$ Then follows from –.

By and , we obtain $$\label{Lem8Eq17}
\left\|u\right\|_{L^{\gamma}{\left(}A_\varepsilon{\left(}R,\lambda_{0,\gamma,\varepsilon}{\right)}{\right)}}\le{\left(}\widetilde{c}_1c_\nu{\right)}^{\frac{1}{\varepsilon}}\max_{{\left(}i_1,\dotsc,i_k{\right)}\in\Phi_{\gamma,\varepsilon}}R^{\tau_{i_1,\dotsc,i_k,\gamma,\varepsilon,\nu}-\sigma_{i_1,\dotsc,i_k,\gamma,\varepsilon}}$$ for all $\nu\in{\left(}0,1{\right)}$, where $\sigma_{i_1,\dotsc,i_k,\gamma,\varepsilon}$ and $\tau_{i_1,\dotsc,i_k,\gamma,\varepsilon,\nu}$ are as in and .

We claim that there exists a constant $\nu_0=\nu_0{\left(}n,\operatorname{\overrightarrow{p}}{\right)}$ such that for any $\nu\in{\left(}0,\nu_0{\right)}$, we have $$\label{Lem8Eq18}
\tau_{i_1,\dotsc,i_k,\gamma,\varepsilon,\nu}-\sigma_{i_1,\dotsc,i_k,\gamma,\varepsilon}\le-\frac{1}{\varepsilon p_\varepsilon}{\left(}1-\frac{n}{\gamma p}{\left(}p_\varepsilon-p{\right)}{\right)}.$$ We prove this claim. By , we obtain $$\begin{aligned}
\label{Lem8Eq19}
\sigma_{i_1,\dotsc,i_k,\gamma,\varepsilon}&=\frac{1}{\varepsilon p_\varepsilon}\bigg(1-\frac{\gamma_{i_1,\dotsc,i_k}}{\gamma}{\left(}\frac{n}{n-p}{\right)}^k+\frac{1}{\gamma}\sum_{j=1}^k{\left(}\frac{n}{n-p}{\right)}^j{\left(}p_\varepsilon-p{\right)}\bigg)\nonumber\\
&=\frac{1}{\varepsilon p_\varepsilon}\bigg(1+\frac{1}{\gamma}{\left(}\frac{n}{n-p}{\right)}^k{\left(}\frac{n}{p}{\left(}p_\varepsilon-p{\right)}-\gamma_{i_1,\dotsc,i_k}{\right)}-\frac{n}{\gamma p}{\left(}p_\varepsilon-p{\right)}\bigg).\end{aligned}$$ In case $p_*-1+\nu\le\gamma_{i_1,\dotsc,i_k}<\frac{n}{p}{\left(}p_\varepsilon-p{\right)}$, since $\tau_{i_1,\dotsc,i_k,\gamma,\varepsilon,\nu}=0$, we deduce directly from . In the remaining case $\gamma_{i_1,\dotsc,i_k}<p_*-1+\nu$, by and , we obtain $$\begin{gathered}
\label{Lem8Eq20}
\tau_{i_1,\dotsc,i_k,\gamma,\varepsilon,\nu}-\sigma_{i_1,\dotsc,i_k,\gamma,\varepsilon}\le-\frac{1}{\varepsilon p_\varepsilon}\bigg(1+\frac{1}{\gamma}{\left(}\frac{n}{n-p}{\right)}^k\bigg(\frac{n}{p}{\left(}p_\varepsilon-p{\right)}-\gamma_{i_1,\dotsc,i_k}\\
-{\left(}1-\frac{\gamma_{i_1,\dotsc,i_k}}{p_*-1+\nu}{\right)}\sum_{i\in\mathcal{I}_1\cup\mathcal{I}_2}\frac{p_\varepsilon-p_i}{p_i}\bigg)-\frac{n}{\gamma p}{\left(}p_\varepsilon-p{\right)}\bigg)\,.\end{gathered}$$ If $\nu$ is small enough so that $p_*-1+\nu<\frac{n}{p}{\left(}p_0-p{\right)}$, i.e. $\nu<\frac{n}{p}{\left(}p_0-p_*{\right)}$, then $$\label{Lem8Eq21}
1-\frac{1}{p_*-1+\nu}\sum_{i\in\mathcal{I}_1\cup\mathcal{I}_2}\frac{p_\varepsilon-p_i}{p_i}<\frac{p}{n{\left(}p_0-p{\right)}}\sum_{i\in{\left(}\mathcal{I}_1\cup\mathcal{I}_2{\right)}^c}\frac{p_0-p_i}{p_i}<0\,,$$ where ${\left(}\mathcal{I}_1\cup\mathcal{I}_2{\right)}^c:=\left\{1,\dotsc,n\right\}\backslash{\left(}\mathcal{I}_1\cup\mathcal{I}_2{\right)}$. It follows from , , and that $$\label{Lem8Eq22}
\tau_{i_1,\dotsc,i_k,\gamma,\varepsilon,\nu}-\sigma_{i_1,\dotsc,i_k,\gamma,\varepsilon}\le-\frac{1}{\varepsilon p_\varepsilon}\bigg(1+\frac{1}{\gamma}{\left(}\frac{n}{n-p}{\right)}^k\varphi_\nu{\left(}p_\varepsilon{\right)}-\frac{n}{\gamma p}{\left(}p_\varepsilon-p{\right)}\bigg)$$ for all $\nu\in(0,\frac{n}{p}{\left(}p_0-p_*{\right)})$, where $$\begin{aligned}
\varphi_\nu{\left(}q{\right)}:&=q-p_--\bigg(1-\frac{\frac{n}{p}{\left(}q-p{\right)}+p_--q}{p_*-1+\nu}\bigg)\underset{i\in\mathcal{I}_1\cup\mathcal{I}_2}{\sum}\frac{q-p_i}{p_i}\\
&=q-p_--\frac{q-p_--\frac{n}{p}{\left(}q-p_*{\right)}}{p_*-1+\nu}\underset{i\in\mathcal{I}_1\cup\mathcal{I}_2}{\sum}\frac{q-p_i}{p_i}\end{aligned}$$ for all $q\in{\mathbb{R}}$. By and by definition of $\overline{p}_0$, we obtain $\varphi_0{\left(}p_0{\right)}>0$. Moreover, it can easily be seen that $\varphi_0{\left(}p_*{\right)}\le0$. Observing that $\varphi_0$ is a quadratic polynomial with positive leading coefficient, we then get that $\varphi_0$ is increasing in ${\left[}p_0,\infty{\right)}$. By continuity of $\varphi_\nu$ with respect to $\nu$, it follows that $\varphi_\nu{\left(}p_\varepsilon{\right)}\le0$ provided that $\nu<\nu_0$ for some constant $\nu_0=\nu_0{\left(}n,\operatorname{\overrightarrow{p}}{\right)}$. By , we then get .

Finally, we fix $\nu=\nu_0/2$, and we obtain by passing to the limit as $\gamma\to\infty$ into and and using the fact that $p_\varepsilon>p_0$ and $R>1$. This ends the proof of Lemma \[Lem8\].

Now, we can conclude the proof of Theorem \[Th6\].

We proceed by contradiction. Suppose that there exists a solution $u$ of such that $$\label{Th6Eq1}
p_0:=\max{\left(}\left\{p_i\in\operatorname{\overrightarrow{p}}\,:\,R_i{\left(}u{\right)}=\infty\right\}{\right)}>\overline{p}_0\,,$$ where $R_i{\left(}u{\right)}$ is as in . Then we can apply Lemma \[Lem8\]. For any $\varepsilon\in{\left(}0,1{\right)}$ and $x\in{\mathbb{R}}^n$, it follows from that $$\label{Th6Eq2}
\left|u{\left(}x{\right)}\right|\le \big(c_2R_\varepsilon{\left(}x{\right)}^{-\frac{1}{p_\varepsilon}}\big)^{\frac{1}{\varepsilon}}\quad\text{where}\quad R_\varepsilon{\left(}x{\right)}:=\sum_{i\in\mathcal{I}_2}\left|x_i\right|^{p_\varepsilon}$$ provided that $\sum_{i\in\mathcal{I}_1}\left|x_i\right|^{\frac{p_\varepsilon p_i}{p_\varepsilon-p_i}}<\frac{5}{4}R_\varepsilon{\left(}x{\right)}^{1/\varepsilon}$ and $A_\varepsilon{\left(}R_\varepsilon{\left(}x{\right)},1/2{\right)}\cap B_{\operatorname{\overrightarrow{p}}}{\left(}0,\max{\left(}r,1{\right)}{\right)}=\emptyset$, where $\mathcal{I}_1$ and $\mathcal{I}_2$ are as in Lemma \[Lem8\], and $r$ is as in Lemma \[Lem3\]. One easily gets that there exists a constant $R_r=R{\left(}n,\operatorname{\overrightarrow{p}},r{\right)}>1$ such that for any $\varepsilon\in{\left(}0,1{\right)}$ and $R>R_r$, we have $A_\varepsilon{\left(}R,1/2{\right)}\cap B_{\operatorname{\overrightarrow{p}}}{\left(}0,\max{\left(}r,1{\right)}{\right)}=\emptyset$. By passing to the limit as $\varepsilon\to0$ into , we then obtain that $u{\left(}x{\right)}=0$ for all $x\in{\mathbb{R}}^n$ such that $$\sum_{i\in\mathcal{I}_2}\left|x_i\right|^{p_0}>\max{\left(}R_r,c_2^{\,p_0}{\right)},$$ and hence $R_i{\left(}u{\right)}<\infty$ for all $i\in\mathcal{I}_2$, which is in contradiction with . This ends the proof of Theorem \[Th6\].

\[Rem2\] As one can see from the above proof, the constant $R_0$ that we obtain in Theorem \[Th6\] depends on $n$, $\operatorname{\overrightarrow{p}}$, $\Lambda$, $\kappa$, $r$, $r_\kappa{\left(}u{\right)}$, and $\left\|u\right\|_{L^{p_*-1,\infty}{\left(}{\mathbb{R}}^n{\right)}}$.

The decay estimates {#Sec7}
===================

In this section, we prove Theorem \[Th1\] in case $p_+<p_*$ and Theorem \[Th7\] below in case $p_*\le p_+<p^*$. The latter implies Theorem \[Th2\] in case $p_+=p_*$ and allows us to complete the proof of Theorem \[Th3\] in case $p_*<p_+<p^*$.

We let $\overline{p}_0$ and $\mathcal{I}_0$ be as in Section \[Sec6\]. We define $q_0$ as the largest real number such that for any $q>q_0$, we have $$\label{Eq17}
\big(q-p_--\frac{n}{p}{\left(}q-p_*{\right)}\big)\sum_{i\in\mathcal{I}_0^c}\frac{q-p_i}{p_i}<{\left(}p_*-1{\right)}{\left(}q-p_-{\right)},$$ where $\mathcal{I}_0^c:=\left\{1,\dotsc,n\right\}\backslash\mathcal{I}_0$. It easily follows from the definition of $\overline{p}_0$ and the fact that $\overline{p}_0<p_+$ in case $p_+>p_*$ that $$\left\{\begin{aligned}
&q_0=\overline{p}_0=p_*&&\text{in case }p_+\le p_*\,,\\
&\overline{p}_0\le q_0<p_+&&\text{in case }p_+>p_*\,.
\end{aligned}\right.$$

We prove the following result.

\[Th7\] Assume that $p_*\le p_+<p^ *$. Let $f:{\mathbb{R}}^n\times{\mathbb{R}}\to{\mathbb{R}}$ be a Caratheodory function such that holds true and $u$ be a solution of . Let $q_0$ be defined as above. Then for any $q>q_0$, there exists a constant $C_q=C{\left(}n,\operatorname{\overrightarrow{p}},\Lambda,u,q{\right)}$ such that $$\label{Th7Eq}
\left|u{\left(}x{\right)}\right|^q+\sum_{i=1}^n\left|\partial_{x_i}u{\left(}x{\right)}\right|^{p_i}\le C_q\bigg(1+\sum_{i\in\mathcal{I}_0^c}\left|x_i\right|^{\frac{qp_i}{q-p_i}}\bigg)^{-1}\quad\text{for a.e. }x\in{\mathbb{R}}^n.$$

We conclude the proofs of Theorems \[Th2\] and \[Th3\] as follows.

In case $p_+=p_*$, since $q_0=\overline{p}_0=p_*$, we get that holds true for all $q>p_*$. Since in this case we have $\mathcal{I}_0^c=\left\{1,\dotsc,n\right\}$, this is exactly the result in Theorem \[Th2\].

In case $p_*<p_+<p^ *$, Points (i) and (ii) in Theorem \[Th3\] follow directly from Theorems \[Th6\] and \[Th7\] and the fact that $\overline{p}_0\le q_0<p_+$.

Now, it remains to prove Theorems \[Th1\] and \[Th7\]. By another application of the iteration scheme in Section \[Sec5\], we prove the following result.

\[Lem9\] Assume that $p_+<p^*$. Let $f:{\mathbb{R}}^n\times{\mathbb{R}}\to{\mathbb{R}}$ be a Caratheodory function such that holds true, $u$ be a solution of , and $\kappa$, $r$, and $K_0$ be as in Lemma \[Lem3\]. Let $q=p_*$ in case $p_+<p_*$ and $q\in{\left(}q_0,p^*{\right)}$ in case $p_*\le p_+<p^*$. For any $\lambda\in{\left(}0,1{\right)}$ and $R>1$, we define $$\label{Lem9Eq1}
A_q{\left(}R,\lambda{\right)}:=\Omega_{{\overrightarrow{q}}}{\left(}\emptyset,1,\mathcal{I}^c_0,R,\lambda{\right)}\quad\text{with}\quad q_i:=\frac{qp_i}{q-p_i}\text{ for all }i\in\mathcal{I}^c_0\,.$$ If $A_q{\left(}R,1/2{\right)}\cap B_{\operatorname{\overrightarrow{p}}}{\left(}0,\max{\left(}r,1{\right)}{\right)}=\emptyset$, then $$\label{Lem9Eq2}
\left\|u\right\|_{L^\infty{\left(}A_q{\left(}R,1/4{\right)}{\right)}}\le c_qR^{-\frac{1}{q}},$$ for some constant $c_q=c\big(n,\operatorname{\overrightarrow{p}},\Lambda,K_0,\left\|u\right\|_{L^{p_*-1,\infty}{\left(}{\mathbb{R}}^n{\right)}},R_0,q\big)$, where $R_0$ is as in Theorem \[Th6\].

By Theorem \[Th6\], we obtain that $A_q{\left(}R,\lambda{\right)}\cap\operatorname{supp}{\left(}u{\right)}$ is bounded. By Lemma \[Lem7\], we then get that for any $\varepsilon\in{\left(}0,1{\right)}$, there exists a constant $c_{q,\varepsilon}=c{\left(}n,\operatorname{\overrightarrow{p}},\Lambda,K_0,q,\varepsilon{\right)}$ such that for any $\gamma>\frac{n}{p}{\left(}\overline{p}_\varepsilon-p{\right)}$, where $\overline{p}_\varepsilon:={\left(}1+\varepsilon{\right)}\overline{p}_0$, we have $$\label{Lem9Eq3}
\left\|u\right\|_{L^{\gamma}{\left(}A_q{\left(}R,\lambda_{0,\gamma,\varepsilon}{\right)}{\right)}}\le c_{q,\varepsilon}\max_{{\left(}i_1,\dotsc,i_k{\right)}\in\Phi_{\gamma,\varepsilon}}\Big(R^{-\sigma_{i_1,\dotsc,i_k,q,\gamma}}\left\|u\right\|_{L^{\gamma_{i_1,\dotsc,i_k}}{\left(}A_q{\left(}R,\lambda_{k,\gamma,\varepsilon}{\right)}{\right)}}^{\frac{\gamma_{i_1,\dotsc,i_k}}{\gamma}{\left(}\frac{n}{n-p}{\right)}^k}\Big),$$ provided that $A_q{\left(}R,1/2{\right)}\cap B_{\operatorname{\overrightarrow{p}}}{\left(}0,\max{\left(}r,1{\right)}{\right)}=\emptyset$, where $\gamma_{i_1,\dotsc,i_k}$ is as in , $\lambda_{k,\gamma,\varepsilon}$ and $\Phi_{\gamma,\varepsilon}$ are as in , and $$\label{Lem9Eq4}
\sigma_{i_1,\dotsc,i_k,q,\gamma}:=\frac{1}{\gamma q}\sum_{j=1}^k{\left(}\frac{n}{n-p}{\right)}^j{\left(}q-p_{i_j}{\right)}.$$

In this case, we follow in large part the same arguments as in the proof of Lemma \[Lem8\]. We set $\varepsilon:={\left(}q-\overline{p}_0{\right)}/\overline{p}_0$ so that $q=\overline{p}_\varepsilon$. Since $q<p^*$ and $\overline{p}_0\ge p_*$, we get $\varepsilon<{\left(}p^*-p_*{\right)}/p_*\le1$. Similarly to , we then obtain that for any ${\left(}i_1,\dotsc,i_k{\right)}\in\Phi_{\gamma,\varepsilon}$ and $\nu\in{\left(}0,1{\right)}$, there exists a constant $c_\nu=c\big(n,\operatorname{\overrightarrow{p}},\Lambda,K_0,\left\|u\right\|_{L^{p_*-1,\infty}{\left(}{\mathbb{R}}^n{\right)}},R_0,\nu\big)$ such that $$\label{Lem9Eq5}
\left\|u\right\|_{L^{\gamma_{i_1,\dotsc,i_k}}{\left(}A_q{\left(}R,\lambda_{k,\gamma,\varepsilon}{\right)}{\right)}}^{\frac{\gamma_{i_1,\dotsc,i_k}}{\gamma}{\left(}\frac{n}{n-p}{\right)}^k}\le c_\nu^{\frac{1}{\varepsilon}}R^{\tau_{i_1,\dotsc,i_k,q,\gamma,\nu}},$$ where $$\label{Lem9Eq6}
\tau_{i_1,\dotsc,i_k,q,\gamma,\nu}:=\max\bigg(0\,,\frac{1}{q\gamma}{\left(}\frac{n}{n-p}{\right)}^k{\left(}1-\frac{\gamma_{i_1,\dotsc,i_k}}{p_*-1+\nu}{\right)}\sum_{i\in\mathcal{I}_0^c}\frac{q-p_i}{p_i}\bigg).$$ It follows from and that $$\label{Lem9Eq7}
\left\|u\right\|_{L^{\gamma}{\left(}A_q{\left(}R,\lambda_{0,\gamma,\varepsilon}{\right)}{\right)}}\le c_{q,\varepsilon}c_\nu^{\frac{1}{\varepsilon}}\max_{{\left(}i_1,\dotsc,i_k{\right)}\in\Phi_{\gamma,\varepsilon}}R^{\tau_{i_1,\dotsc,i_k,q,\gamma,\nu}-\sigma_{i_1,\dotsc,i_k,q,\gamma}}$$ for all $\nu\in{\left(}0,1{\right)}$, where $\sigma_{i_1,\dotsc,i_k,q,\gamma}$ and $\tau_{i_1,\dotsc,i_k,q,\gamma,\nu}$ are as in and .

In the same way as in the proof of , we then obtain $$\label{Lem9Eq8}
\tau_{i_1,\dotsc,i_k,q,\gamma,\nu}-\sigma_{i_1,\dotsc,i_k,q,\gamma}\le-\frac{1}{q}{\left(}1-\frac{n}{\gamma p}{\left(}q-p{\right)}{\right)}$$ provided that $$\label{Lem9Eq9}
q-p_--\frac{q-p_--\frac{n}{p}{\left(}q-p_*{\right)}+\nu}{p_*-1+\nu}\sum_{i\in\mathcal{I}_0^c}\frac{q-p_i}{p_i}>0\,.$$ By , we get that holds true provided that $\nu<\nu_0$ for some constant $\nu_0=\nu_0{\left(}n,\operatorname{\overrightarrow{p}}{\right)}$.

Finally, we fix $\nu=\nu_q/2$, and we obtain by passing to the limit as $\gamma\to\infty$ into and . This ends the proof of Lemma \[Lem9\].

In this case, we have $\overline{p}_0=p^*$ and $\mathcal{I}_0^c=\left\{1,\dotsc,n\right\}$. We claim that there exists a constant $\varepsilon_0=\varepsilon_0{\left(}n,\operatorname{\overrightarrow{p}}{\right)}\in{\left(}0,1{\right)}$ such that for any $\varepsilon\in{\left(}0,\varepsilon_0{\right)}$ and ${\left(}i_1,\dotsc,i_k{\right)}\in\Phi_{\gamma,\varepsilon}$, we have $$\label{Lem9Eq10}
\left\|u\right\|_{L^{\gamma_{i_1,\dotsc,i_k}}{\left(}A_q{\left(}R,\lambda_{k,\gamma,\varepsilon}{\right)}{\right)}}^{\frac{\gamma_{i_1,\dotsc,i_k}}{\gamma}{\left(}\frac{n}{n-p}{\right)}^k}\le c_\varepsilon R^{\tau_{i_1,\dotsc,i_k,p_*,\gamma}}$$ for some constant $c_\varepsilon=c\big(n,\operatorname{\overrightarrow{p}},\Lambda,K_0,\left\|u\right\|_{L^{p_*-1,\infty}{\left(}{\mathbb{R}}^n{\right)}},\varepsilon\big)$, where $$\label{Lem9Eq11}
\tau_{i_1,\dotsc,i_k,p_*,\gamma}:=\frac{p_*-1-\gamma_{i_1,\dotsc,i_k}}{p_*\gamma}{\left(}\frac{n}{n-p}{\right)}^k.$$ We assume that ${\left(}1-\varepsilon{\right)}p_*>p$, i.e. $\varepsilon<\frac{p-1}{n-1}$, and we separate two cases:

-   Case 1: $\gamma_{i_1,\dotsc,i_k}\le\frac{n}{p}{\left(}{\left(}1-\varepsilon{\right)}p_*-p{\right)}$,

-   Case 2: $\frac{n}{p}{\left(}{\left(}1-\varepsilon{\right)}p_*-p{\right)}<\gamma_{i_1,\dotsc,i_k}<\frac{n}{p}{\left(}{\left(}1+\varepsilon{\right)}p_*-p{\right)}$.

We begin with proving in Case 1. By a generalized version of Hölder’s inequality (see for instance Grafakos [@Gra]\*[Exercise 1.1.11]{}), we obtain $$\begin{gathered}
\label{Lem9Eq12}
\left\|u\right\|^{\gamma_{i_1,\dotsc,i_k}}_{L^{\gamma_{i_1,\dotsc,i_k}}{\left(}A_q{\left(}R,\lambda_{k,\gamma,\varepsilon}{\right)}{\right)}}\le\frac{p_*-1}{p_*-1-\gamma_{i_1,\dotsc,i_k}}\left|A_q{\left(}R,\lambda_{k,\gamma,\varepsilon}{\right)}\right|^{1-\frac{\gamma_{i_1,\dotsc,i_k}}{p_*-1}}\left\|u\right\|^{\gamma_{i_1,\dotsc,i_k}}_{L^{p_*-1,\infty}{\left(}A_q{\left(}R,\lambda_{k,\gamma,\varepsilon}{\right)}{\right)}}.\end{gathered}$$ Direct computations give $$\label{Lem9Eq13}
\left|A_q{\left(}R,\lambda_{k,\gamma,\varepsilon}{\right)}\right|\le CR^{\overset{n}{\underset{i=1}{\sum}}\frac{p_*-p_i}{p_*p_i}}=CR^{\frac{p_*-1}{p_*}}$$ for some constant $C=C{\left(}n,\operatorname{\overrightarrow{p}}{\right)}$. Since $\gamma_{i_1,\dotsc,i_k}\le\frac{n}{p}{\left(}{\left(}1-\varepsilon{\right)}p_*-p{\right)}$ and $u\in L^{p_*-1,\infty}{\left(}{\mathbb{R}}^n{\right)}$, it follows from and that $$\label{Lem9Eq14}
\left\|u\right\|^{\gamma_{i_1,\dotsc,i_k}}_{L^{\gamma_{i_1,\dotsc,i_k}}{\left(}A_q{\left(}R,\lambda_{k,\gamma,\varepsilon}{\right)}{\right)}}\le C\varepsilon^{-1}R^{\frac{p_*-1-\gamma_{i_1,\dotsc,i_k}}{p_*}}$$ for some constant $C=C\big(n,\operatorname{\overrightarrow{p}},\left\|u\right\|_{L^{p_*-1,\infty}{\left(}{\mathbb{R}}^n{\right)}}\big)$. Moreover, since $k\le k^+_{\gamma,\varepsilon}$, by , we get $$\label{Lem9Eq15}
\frac{1}{\gamma}{\left(}\frac{n}{n-p}{\right)}^k<\frac{1}{\varepsilon{\left(}n-1{\right)}}\,.$$ Then follows from and .

Now, suppose that we are in case 2. By interpolation, we obtain $$\label{Lem9Eq16}
\left\|u\right\|_{L^{\gamma_{i_1,\dotsc,i_k}}{\left(}A_q{\left(}R,\lambda_{k,\gamma,\varepsilon}{\right)}{\right)}}\le\left\|u\right\|^\theta_{L^{\frac{n}{p}{\left(}{\left(}1-\varepsilon{\right)}p_*-p{\right)}}{\left(}A_q{\left(}R,\lambda_{k,\gamma,\varepsilon}{\right)}{\right)}}\left\|u\right\|^{1-\theta}_{L^{\frac{n}{p}{\left(}{\left(}1+\varepsilon{\right)}p_*-p{\right)}}{\left(}A_q{\left(}R,\lambda_{k,\gamma,\varepsilon}{\right)}{\right)}},$$ where $\theta\in{\left(}0,1{\right)}$ is such that $$\label{Lem9Eq17}
\frac{\theta}{\frac{n}{p}{\left(}{\left(}1-\varepsilon{\right)}p_*-p{\right)}}+\frac{1-\theta}{\frac{n}{p}{\left(}{\left(}1+\varepsilon{\right)}p_*-p{\right)}}=\frac{1}{\gamma_{i_1,\dotsc,i_k}}\,.$$ Similarly to , we get $$\label{Lem9Eq18}
\left\|u\right\|^{\frac{n}{p}{\left(}{\left(}1-\varepsilon{\right)}p_*-p{\right)}}_{L^{\frac{n}{p}{\left(}{\left(}1-\varepsilon{\right)}p_*-p{\right)}}{\left(}A_q{\left(}R,\lambda_{k,\gamma,\varepsilon}{\right)}{\right)}}\le C\varepsilon^{-1}R^{\frac{n}{p}\varepsilon}$$ for some constant $C=C\big(n,\operatorname{\overrightarrow{p}},K_0,\left\|u\right\|_{L^{p_*-1,\infty}{\left(}{\mathbb{R}}^n{\right)}}\big)$. On the other hand, Lemma \[Lem4\] gives $$\label{Lem9Eq19}
\left\|u\right\|^{\frac{n}{p}{\left(}{\left(}1+\varepsilon{\right)}p_*-p{\right)}}_{L^{\frac{n}{p}{\left(}{\left(}1+\varepsilon{\right)}p_*-p{\right)}}{\left(}A_q{\left(}R,\lambda_{k,\gamma,\varepsilon}{\right)}{\right)}}\le C\max_{i=1,\dotsc,n}\Big(\varepsilon^{-p_i}R^{\frac{p_i-p_*}{p_*}}\left\|u\right\|_{L^{\varepsilon{\left(}n-1{\right)}+p_i-1}{\left(}A_q{\left(}R,1/2{\right)}{\right)}}^{\varepsilon{\left(}n-1{\right)}+p_i-1}\Big)^{\frac{n}{n-p}}$$ for some constant $C=C{\left(}n,\operatorname{\overrightarrow{p}},\Lambda,K_0{\right)}$. We define $\varepsilon_0:={\left(}p_*-p_+{\right)}/{\left(}p_*+2n-2{\right)}$ so that for any $\varepsilon\in{\left(}0,\varepsilon_0{\right)}$ and $i=1,\dotsc,n$, we have $$\varepsilon{\left(}n-1{\right)}+p_i-1<\frac{n}{p}{\left(}{\left(}1-\varepsilon{\right)}p_*-p{\right)}.$$ Similarly to , we then get $$\label{Lem9Eq20}
\left\|u\right\|^{\varepsilon{\left(}n-1{\right)}+p_i-1}_{L^{\varepsilon{\left(}n-1{\right)}+p_i-1}{\left(}A_q{\left(}R,\lambda_{k+1,\gamma,\varepsilon}{\right)}{\right)}}\le CR^{\frac{p_*-p_i-\varepsilon{\left(}n-1{\right)}}{p_*}}$$ for some constant $C=C\big(n,\operatorname{\overrightarrow{p}},K_0,\left\|u\right\|_{L^{p_*-1,\infty}{\left(}{\mathbb{R}}^n{\right)}}\big)$. By putting together –, we obtain $$\label{Lem9Eq21}
\left\|u\right\|^{\gamma_{i_1,\dotsc,i_k}}_{L^{\gamma_{i_1,\dotsc,i_k}}{\left(}A_q{\left(}R,\lambda_{k,\gamma,\varepsilon}{\right)}{\right)}}\le C\varepsilon^{-s}R^{\frac{p_*-1-\gamma_{i_1,\dotsc,i_k}}{p_*}}$$ for some constants $C=C\big(n,\operatorname{\overrightarrow{p}},\Lambda,K_0,\left\|u\right\|_{L^{p_*-1,\infty}{\left(}{\mathbb{R}}^n{\right)}}\big)$ and $s=s{\left(}n,\operatorname{\overrightarrow{p}}{\right)}>0$. Then follows from and .

By and , we obtain that for any $\varepsilon\in{\left(}0,\varepsilon_0{\right)}$, there exists a constant $\widetilde{c}_\varepsilon=c\big(n,\operatorname{\overrightarrow{p}},\Lambda,K_0,\left\|u\right\|_{L^{p_*-1,\infty}{\left(}{\mathbb{R}}^n{\right)}},\varepsilon\big)$ such that $$\label{Lem9Eq22}
\left\|u\right\|_{L^\gamma{\left(}A_q{\left(}R,\lambda_{0,\gamma,\varepsilon}{\right)}{\right)}}\le \widetilde{c}_\varepsilon\max_{{\left(}i_1,\dotsc,i_k{\right)}\in\Phi_{\gamma,\varepsilon}}R^{\tau_{i_1,\dotsc,i_k,p_*,\gamma}-\sigma_{i_1,\dotsc,i_k,p_*,\gamma}},$$ where $\sigma_{i_1,\dotsc,i_k,p_*,\gamma}$ and $\tau_{i_1,\dotsc,i_k,p_*,\gamma}$ are as in and .

From , we derive $$\sigma_{i_1,\dotsc,i_k,p_*,\gamma}=\frac{1}{p_*}\bigg(1+\frac{1}{\gamma}{\left(}\frac{n}{n-p}{\right)}^k{\left(}p_*-1-\gamma_{i_1,\dotsc,i_k}{\right)}-\frac{p_*-1}{\gamma}\bigg),$$ and hence $$\label{Lem9Eq23}
\tau_{i_1,\dotsc,i_k,p_*,\gamma}-\sigma_{i_1,\dotsc,i_k,p_*,\gamma}=-\frac{1}{p_*}\bigg(1-\frac{p_*-1}{\gamma}\bigg).$$ Finally, we fix $\varepsilon=\varepsilon_0/2$, and we obtain by passing to the limit as $\gamma\to\infty$ into and . This ends the proof of Lemma \[Lem9\] in case $p_+<p_*$ and $q=p_*$.

Now, we can prove Theorems \[Th7\] and \[Th1\].

As is easily seen, it is sufficient to prove for $q\in{\left(}q_0,p^*{\right)}$. Let $u$ be a solution of and $q>q_0$. We define $$u_R{\left(}y{\right)}:=R^{\frac{1}{q}}\cdot u{\left(}\tau_R{\left(}y{\right)}{\right)},\quad\text{where}\quad\tau_R{\left(}y{\right)}:=\big(R^{\frac{q-p_1}{qp_1}}y_1,\dotsc,R^{\frac{q-p_n}{qp_n}}y_n\big)$$ for all $R>1$ and $y\in{\mathbb{R}}^n$. By Lemma \[Lem9\], we obtain $$\label{Th1Eq13}
\left\|u_R\right\|_{L^\infty{\left(}A_q{\left(}1,1/4{\right)}{\right)}}\le c_q\,.$$ provided that $A_q{\left(}R,1/2{\right)}\cap B_{\operatorname{\overrightarrow{p}}}{\left(}0,\max{\left(}r,1{\right)}{\right)}=\emptyset$, where $r$ be as in Lemma \[Lem3\]. One easily gets the existence of a constant $R_r=R{\left(}n,\operatorname{\overrightarrow{p}},r{\right)}>1$ such that $A_q{\left(}R,1/2{\right)}\cap B_{\operatorname{\overrightarrow{p}}}{\left(}0,\max{\left(}r,1{\right)}{\right)}=\emptyset$ for all $R>R_r$. Moreover, by , we obtain $$\label{Th1Eq14}
-\Delta_{\operatorname{\overrightarrow{p}}}u_R=R^{\frac{q-1}{q}}\cdot f\big(\tau_R{\left(}y{\right)},R^{-\frac{1}{q}}\cdot u_R\big)\quad\text{in }{\mathbb{R}}^n,$$ and gives $$\label{Th1Eq15}
\big|R^{\frac{q-1}{q}}\cdot f\big(\tau_R{\left(}y{\right)},R^{-\frac{1}{q}}\cdot u_R\big)\big|\le\Lambda\cdot R^{\frac{q-p^*}{q}}\cdot\left|u_R\right|^{p^*-1}.$$ Since $q-p^*\le0$, by – and Lieberman’s gradient estimates [@Lie], we get that there exists a constant $c'_q=c{\left(}n,\operatorname{\overrightarrow{p}},\Lambda,c_q{\right)}$ such that $$\label{Th1Eq16}
\left\|\nabla u_R\right\|_{L^\infty{\left(}A_q{\left(}1,1/8{\right)}{\right)}}\le c'_q\,.$$ For any $x\in{\mathbb{R}}^n$, it follows from and that $$\left|u{\left(}x{\right)}\right|^q+\sum_{i=1}^n\left|\partial_{x_i}u{\left(}x{\right)}\right|^{p_i}\le c''_qR{\left(}x{\right)}^{-1}\quad\text{where}\quad R{\left(}x{\right)}:=\sum_{i\in\mathcal{I}_0^c}\left|x_i\right|^{\frac{qp_i}{q-p_i}}$$ for some constant $c''_q=c{\left(}n,\operatorname{\overrightarrow{p}},\Lambda,c_q{\right)}$, provided that $R{\left(}x{\right)}>R_r$. This ends the proof of Theorem \[Th7\].

We fix $q=p^*$ in this case and we follow the same arguments as in the above proof of Theorem \[Th7\].

\[Rem3\] As one can see from the above proofs, the constants $C_0$ and $C_q$ that we obtain in and  depend on $n$, $\operatorname{\overrightarrow{p}}$, $q$, $\Lambda$, $\kappa$, $r$, $r_\kappa{\left(}u{\right)}$, $\left\|u\right\|_{L^{p_*-1,\infty}{\left(}{\mathbb{R}}^n{\right)}}$, and $\left\|u\right\|_{W^{1,\infty}{\left(}{\mathbb{R}}^n\backslash\Omega_r{\right)}}$, where $\Omega_r:=\big\{x\in{\mathbb{R}}^n:\,\sum_{i=1}^n\left|x_i\right|^{\frac{qp_i}{q-p_i}}>R_r\big\}$ for some constant $R_r=R{\left(}n,\operatorname{\overrightarrow{p}},r{\right)}$.

Kato-type inequality {#App}
====================

In this section, we prove a weak version of Kato’s inequality [@Kato] for the operator $\Delta_{\operatorname{\overrightarrow{p}}}$. This result is used in Sections \[Sec3\] and \[Sec4\]. A similar result has been proven by Cuesta Leon [@Cue] in the context of the $p$–Laplace operator.

For any $f\in L^1_{\operatorname{loc}}{\left(}{\mathbb{R}}^n{\right)}$, we say that a function $u\in D^{1,\operatorname{\overrightarrow{p}}}{\left(}{\mathbb{R}}^n{\right)}$ is a solution of the inequality $$-\Delta_{\operatorname{\overrightarrow{p}}}u\le f\quad\text{in }{\mathbb{R}}^n$$ if we have $$\sum_{i=1}^n\int_{{\mathbb{R}}^n}\left|\partial_{x_i}u\right|^{p_i-2}{\left(}\partial_{x_i}u{\right)}{\left(}\partial_{x_i}\varphi{\right)}dx\le\int_{{\mathbb{R}}^n}f\varphi\,dx$$ for all nonnegative, smooth function $\varphi$ with compact support in ${\mathbb{R}}^n$.

We state our result as follows.

\[Pr2\] Let $f_1,f_2\in L^1_{\operatorname{loc}}{\left(}{\mathbb{R}}^n{\right)}$ and $u_1,u_2\in D^{1,\operatorname{\overrightarrow{p}}}{\left(}{\mathbb{R}}^n{\right)}$ be solutions of the inequalities $$\label{Pr2Eq1}
-\Delta_{\operatorname{\overrightarrow{p}}}u_j\le f_j\quad\text{in }{\mathbb{R}}^n$$ for $j=1,2$. Then the function $u:=\max{\left(}u_1,u_2{\right)}$ is a solution of the inequality $$\label{Pr2Eq2}
-\Delta_{\operatorname{\overrightarrow{p}}}u\le f\quad\text{in }{\mathbb{R}}^n,$$ where $f{\left(}x{\right)}:=f_1{\left(}x{\right)}$ if $u_1{\left(}x{\right)}>u_2{\left(}x{\right)}$, $f{\left(}x{\right)}:=f_2{\left(}x{\right)}$ if $u_1{\left(}x{\right)}\le u_2{\left(}x{\right)}$ for all $x\in{\mathbb{R}}^n$.

We essentially follow the lines of Cuesta Leon [@Cue]\*[Proposition 3.2]{}. For any $\varepsilon>0$ and $x\in{\mathbb{R}}^n$, we define $$\eta_{1,\varepsilon}{\left(}x{\right)}:=\eta_\varepsilon{\left(}u_1{\left(}x{\right)}-u_2{\left(}x{\right)}{\right)}\quad\text{and}\quad\eta_{2,\varepsilon}{\left(}x{\right)}:=1-\eta_{1,\varepsilon}{\left(}x{\right)},$$ where $\eta_\varepsilon\in C^1{\left(}R{\right)}$ is such that $\eta_\varepsilon\equiv0$ in ${\left(}-\infty,0{\right]}$, $\eta_\varepsilon\equiv1$ in ${\left[}1,\infty{\right)}$, $0\le\eta_\varepsilon\le1$ and $\eta'_\varepsilon\ge0$ in ${\left(}0,1{\right)}$. In particular, for $j=1,2$, we have $\eta_{j,\varepsilon}\in D^{1,\operatorname{\overrightarrow{p}}}{\left(}{\mathbb{R}}^n{\right)}$, $0\le\eta_{j,\varepsilon}\le1$ in ${\mathbb{R}}^n$, and $$\label{Pr2Eq3}
\eta_{j,\varepsilon}{\left(}x{\right)}\longrightarrow\left\{\begin{aligned}&1&&\text{if }x\in\Omega_j\\&0&&\text{if }x\in{\mathbb{R}}^n\backslash\Omega_j\end{aligned}\right.$$ as $\varepsilon\to0$ for all $x\in{\mathbb{R}}^n$, where $\Omega_1:=\left\{x\in{\mathbb{R}}^n:\,u_1{\left(}x{\right)}>u_2{\left(}x{\right)}\right\}$ and $\Omega_2:={\mathbb{R}}^n\backslash\Omega_1$. For any nonnegative, smooth function $\varphi$ with compact support in ${\mathbb{R}}^n$, testing with $\varphi\eta_{j,\varepsilon}$ gives $$\begin{gathered}
\label{Pr2Eq4}
\sum_{i=1}^n\int_{{\mathbb{R}}^n}\left|\partial_{x_i}u_j\right|^ {p_i-2}{\left(}\partial_{x_i}u_j{\right)}{\left(}\partial_{x_i}\varphi{\right)}\eta_{j,\varepsilon}dx\\
+{\left(}-1{\right)}^{j-1}\sum_{i=1}^n\int_{{\mathbb{R}}^n}\left|\partial_{x_i}u_j\right|^ {p_i-2}{\left(}\partial_{x_i}u_j{\right)}{\left(}\partial_{x_i}u_1-\partial_{x_i}u_2{\right)}\eta'_\varepsilon{\left(}u_1-u_2{\right)}\varphi\,dx\le\int_{{\mathbb{R}}^n}f_j\varphi\eta_{j,\varepsilon}dx\,.\end{gathered}$$ By and since $f_j\in L^1_{\operatorname{loc}}{\left(}{\mathbb{R}}^n{\right)}$ and $u_j\in D^{1,\operatorname{\overrightarrow{p}}}{\left(}{\mathbb{R}}^n{\right)}$, we obtain $$\begin{aligned}
\int_{{\mathbb{R}}^n}f_j\varphi\eta_{j,\varepsilon}dx&\longrightarrow\int_{\Omega_j}f_j\varphi\,dx\,,\label{Pr2Eq5}\\
\int_{{\mathbb{R}}^n}\left|\partial_{x_i}u_j\right|^ {p_i-2}{\left(}\partial_{x_i}u_j{\right)}{\left(}\partial_{x_i}\varphi{\right)}\eta_{j,\varepsilon}dx&\longrightarrow\int_{\Omega_j}\left|\partial_{x_i}u_j\right|^ {p_i-2}{\left(}\partial_{x_i}u_j{\right)}{\left(}\partial_{x_i}\varphi{\right)}dx\label{Pr2Eq6}\end{aligned}$$ as $\varepsilon\to0$ for all $i=1,\dotsc,n$. Moreover, since $\eta'_\varepsilon\ge0$ and $\varphi\ge0$, we get $$\label{Pr2Eq7}
\sum_{i=1}^n\int_{{\mathbb{R}}^n}{\left(}\left|\partial_{x_i}u_1\right|^ {p_i-2}\partial_{x_i}u_1-\left|\partial_{x_i}u_2\right|^ {p_i-2}\partial_{x_i}u_2{\right)}{\left(}\partial_{x_i}u_1-\partial_{x_i}u_2{\right)}\eta'_\varepsilon{\left(}u_1-u_2{\right)}\varphi\,dx\ge0\,.$$ It follows from – that $$\label{Pr2Eq8}
\sum_{j=1}^2\sum_{i=1}^n\int_{\Omega_j}\left|\partial_{x_i}u_j\right|^ {p_i-2}{\left(}\partial_{x_i}u_j{\right)}{\left(}\partial_{x_i}\varphi{\right)}dx\le\sum_{j=1}^2\int_{\Omega_j}f_j\varphi\,dx$$ and hence holds true since $u\equiv u_j$ and $f\equiv f_j$ on $\Omega_j$ for $j=1,2$.

[99]{}

[^1]: To appear in [*Advances in Mathematics*]{}
