---
abstract: |
    The aim of this paper is to prove multiplicity of solutions for nonlocal fractional equations modeled by $$\left\{
    \begin{array}{ll}
    (-\Delta)^s u-\lambda u=f(x,u) & {\mbox{ in }} \Omega\\
    u=0 & {\mbox{ in }} {\mathbb R}^n\setminus \Omega\,,
    \end{array} \right.$$ where $s\in (0,1)$ is fixed, $(-\Delta)^s$ is the fractional Laplace operator, $\lambda$ is a real parameter, $\Omega\subset {\mathbb R}^n$, $n>2s$, is an open bounded set with continuous boundary and nonlinearity $f$ satisfies natural superlinear and subcritical growth assumptions. Precisely, along the paper we prove the existence of at least three non-trivial solutions for this problem in a suitable left neighborhood of any eigenvalue of $(-\Delta)^s$. At this purpose we employ a variational theorem of mixed type (one of the so-called $\nabla$-theorems).
address:
- 'Dipartimento PAU, Università ‘Mediterranea’ di Reggio Calabria, Via Melissari 24, 89124 Reggio Calabria, Italy'
- 'Dipartimento di Matematica e Informatica, Università degli Studi di Perugia, Via Vanvitelli, 1 06123 Perugia, Italy'
- 'Dipartimento di Scienze di Base e Fondamenti (DiSBeF), Università degli Studi di Urbino ‘Carlo Bo’, Piazza della Repubblica, 13 61029 Urbino (Pesaro e Urbino) ITALY '
author:
- Giovanni Molica Bisci
- Dimitri Mugnai
- Raffaella Servadei
title: 'On multiple solutions for nonlocal fractional problems via $\nabla$-theorems'
---

[^1]

Introduction {#sec:introduzione}
============

Classical critical point theorems, like the Mountain Pass Theorem, the Linking Theorem or the Saddle Point Theorem (see [@ar; @puccirad; @rabinowitz; @struwe]), have been extensively used in order to construct non-trivial solutions for nonlocal equations of the type $$\left\{
\begin{array}{ll}
(-\Delta)^s u=f(x,u) & {\mbox{ in }} \Omega\\
u=0 & {\mbox{ in }} {\mathbb R}^n\setminus \Omega\,
\end{array} \right.$$ under different growth assumptions on $f$ (see, e.g., [@colorado; @barrioscoloradoservadei; @capella; @fiscella; @fsvBS; @molicaservadei; @sY; @sBNRES; @svmountain; @svlinking; @servadeivaldinociBN; @servadeivaldinociBNLOW; @servadeivaldinociCFP; @tan] and references therein, and [@cabretan] for a minimization procedure). Here, $s\in (0,1)$ is fixed and $(-\Delta)^s$ is the fractional Laplace operator, which (up to normalization factors) may be defined as $$\label{2}
-(-\Delta)^s u(x)=
\int_{{\mathbb R}^n}\frac{u(x+y)+u(x-y)-2u(x)}{|y|^{n+2s}}\,dy\,,
\,\,\,\,\, x\in {\mathbb R}^n\,.$$

The aim of this paper is to focus on the existence of multiple solutions for this kind of problems, in the case when $f$ is a superlinear and subcritical nonlinearity. Precisely, we will study the following problem: $$\label{problemaK}
\left\{
\begin{array}{ll}
\mathcal L_K u+\lambda u+f(x,u)=0 & {\mbox{ in }} \Omega\\
u=0 & {\mbox{ in }} {\mathbb R}^n\setminus \Omega.
\end{array} \right.$$ Here $s\in (0,1)$ is fixed, $n>2s$, $\Omega\subset {\mathbb R}^n$ is an open bounded set with continuous boundary, and the non-local operator $\mathcal L_K$ is defined as $$\label{lk}
\mathcal L_Ku(x)=
\int_{{\mathbb R}^n}\Big(u(x+y)+u(x-y)-2u(x)\Big)K(y)\,dy\,,
\,\,\,\,\, x\in {\mathbb R}^n,$$ and $K:{\mathbb R}^n\setminus\{0\}\rightarrow(0,+\infty)$ is a function with the properties that $$\label{kernel}
{\mbox{$m K\in L^1({\mathbb R}^n)$, where $m(x)=\min \{|x|^2, 1\}$\,;}}$$ $$\label{kernelfrac}
\mbox{there exists}\,\, \theta>0\,\,
\mbox{such that}\,\, K(x)\geq \theta |x|^{-(n+2s)}\,\,
\mbox{for any}\,\, x\in {\mathbb R}^n \setminus\{0\}\,.\\$$ A model for $K$ is given by $K(x)=|x|^{-(n+2s)}$. In this case $\mathcal L_K$ is the fractional Laplace operator $-(-\Delta)^s$.

In the recent papers [@fiscellamolicaservadeiCFS; @152; @molicaMRL; @servadeiKavian] the multiplicity of solutions in the nonlocal fractional setting has been addressed by means of classical critical point theorems in the spirit of the ones cited above.

In [@marsac:sns] (see also [@MMu; @marsac]) Marino and Saccon introduced new critical point theorems, the so-called *theorems of mixed type*, or *$\nabla$-theorems*, which allow to get multiplicity results for semilinear elliptic problems (see, for instance, [@mms; @mug4; @mugnaiNODEA; @Mug; @mugpag; @OL; @fwang; @fwang2; @wzz]).

We think that a natural question is whether or not these techniques may be adapted to the fractional Laplacian setting. One can define a fractional power of the Laplacian using its spectral decomposition: indeed, in [@mugpag] the same problem considered along this paper, but for this spectral fractional Laplacian, has been considered. As in [@mugpag], the purpose of this paper is to investigate the existence of multiple weak solutions for .

A weak solution of is a solution of the following problem: $$\label{problemaKweak}
\left\{\begin{array}{l}
{\displaystyle \int_{{\mathbb R}^n \times {\mathbb R}^n } (u(x)-u(y))(\varphi(x)-\varphi(y))K(x-y) dx\,dy-\lambda \int_\Omega u(x)\varphi(x)\,dx}\\
\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad
{\displaystyle = \int_\Omega f(x,u(x))\varphi(x)dx}\,\,\,\,\,\,\, \forall\,\, \varphi \in X_0\\
\\
u\in X_0
\end{array}\right.$$ (for this see [@sv Lemma 5.6] and [@servadeivaldinociBN footnote 3]). Here $X_0$ is defined as follows: $X$ is the linear space of Lebesgue measurable functions from ${\mathbb R}^n$ to ${\mathbb R}$ such that the restriction to $\Omega$ of any function $g$ in $X$ belongs to $L^2(\Omega)$ and $$\mbox{the map}\,\,\,
(x,y)\mapsto (g(x)-g(y))\sqrt{K(x-y)}\,\,\, \mbox{is in}\,\,\,
L^2\big({\mathbb R}^n \times {\mathbb R}^n \setminus ({\mathcal C}\Omega\times
{\mathcal C}\Omega), dxdy\big)\,,$$ where ${\mathcal C}\Omega:={\mathbb R}^n \setminus\Omega$. Moreover, $$X_0=\{g\in X : g=0\,\, \mbox{a.e. in}\,\,
{\mathbb R}^n\setminus
\Omega\}\,.$$

As we said here above, we suppose that equation  is superlinear and subcritical, that is its right-hand side $f:\Omega\times {\mathbb R}\to {\mathbb R}$ verifies the following conditions: $$\label{caratheodory}
f\,\, \mbox{is a Carath\'eodory function};$$ $$\label{crescita}
\begin{aligned}
&\mbox{there exist}\,\, a_1, a_2>0\,\,\mbox{and}\,\, q\in (2, 2^*), 2^*=2n/(n-2s)\,,\,\, \mbox{such that}\\
&\qquad \qquad |f(x,t)|\le a_1+a_2|t|^{q-1}\,\, \mbox{a.e.}\,\, x\in \Omega, t\in {\mathbb R}\,;
\end{aligned}$$ $$\label{a3a4}
\begin{aligned}
\mbox{there exist}\,\, & \mbox{two positive constants}\,\, a_3\,\,
\mbox{and}\,\, a_4\,\, \mbox{such that} \\
& F(x,t)\geq a_3|t|^q-a_4\,\, \mbox{a.e.}\,\, x\in \Omega, t\in {\mathbb R};
\end{aligned}$$ $$\label{cond0}
{\displaystyle \lim_{|t|\to 0}\frac{f(x,t)}{|t|}=0}\,\, \mbox{uniformly in}\,\, x\in \Omega\,;$$ $$\label{mu0}
0<q F(x,t)\le tf(x,t)\,\,\mbox{a.e.}\,\, x\in \Omega, t\in {\mathbb R}\setminus\{0\},$$ where $q$ is given in and the function $F$ is the primitive of $f$ with respect to the second variable, that is $$\label{F}
{\displaystyle F(x,t):=\int_0^t f(x,\tau)d\tau}\,.$$

As a model for $f$ we can take the function $f(x,t)=a(x)|t|^{q-2}t$, with $a\in L^\infty(\Omega)$, $\inf_\Omega a>0$ and $q\in (2, 2^*)$. The exponent $2^*$ here plays the role of a fractional critical Sobolev exponent (see, e.g. [@valpal Theorem 6.5]).

As remarked in [@dmnodea2], condition is not a mere consequence of , and must be assumed [*a priori*]{}, unless $f:\bar \Omega \times {\mathbb R}\to {\mathbb R}$ is continuous and holds for every $(x,t)\in \bar \Omega \times {\mathbb R}$.

The main result of the present paper can be stated as follows:

\[thmain\] Let $s\in (0,1)$, $n>2s$ and $\Omega$ be an open bounded subset of ${\mathbb R}^n$ with continuous boundary. Let $K:{\mathbb R}^n\setminus\{0\}\rightarrow(0,+\infty)$ be a function satisfying and and let $f:\Omega\times {\mathbb R}\to {\mathbb R}$ be a function verifying –.

Then, for every eigenvalue $\lambda_k$ of $-\mathcal L_K$ with homogeneous Dirichlet boundary data, there exists a left neighborhood $\mathcal O_k$ of $\lambda_k$ such that problem  admits at least three non-trivial weak solutions for all $\lambda\in \mathcal O_k$.

In the non-local framework, the simplest example we can deal with is given by the fractional Laplacian, according to the following result:

\[thlapfrac\] Let $s\in (0,1)$, $n>2s$ and $\Omega$ be an open bounded subset of ${\mathbb R}^n$ with continuous boundary. If $f:\Omega\times {\mathbb R}\to {\mathbb R}$ is a function verifying –, then the problem $$\label{problema}
\left\{
\begin{array}{ll}
(-\Delta)^s u-\lambda u=f(x,u) & {\mbox{ in }} \Omega\\
u=0 & {\mbox{ in }} {\mathbb R}^n\setminus \Omega
\end{array} \right.$$ admits at least three non-trivial weak solutions belonging to $H^s({\mathbb R}^n)$ in a suitable left neighborhood of any eigenvalue of $(-\Delta)^s$ with homogeneous Dirichlet boundary data.

When $s=1$, problem  reduces to a standard semilinear Laplace equation: in this sense Theorem \[thlapfrac\] may be seen as the fractional version of the result in [@mugnaiNODEA Theorem 1].

We prove Theorem \[thmain\] employing variational and topological methods. Precisely, we apply a $\nabla$-theorem due to Marino and Saccon in [@marsac:sns], see Theorem \[thmarinosaccon\] below. The main difficulties in applying such a theorem are obviously related to the nonlocal nature of the problem.

The paper is organized as follows. In Section \[sec:preliminary\] we collect the notation and some preliminary observations. In Section \[sec:compactness\] we discuss the compactness property of the Euler-Lagrange functional associated with the problem under consideration, while Section \[sec:geometry\] is devoted to its geometric structure. In Section \[sec:nablacondition\] we prove the $\nabla$-condition, which is one of the main ingredient of the critical point theorem we employ in order to get our multiplicity result. Finally, in Section \[sec:proofthmain\] we prove Theorem \[thmain\].

Preliminaries {#sec:preliminary}
=============

In this section we give some preliminary results.

Variational setting {#subsec:variational}
-------------------

First of all, we need some notations. In the sequel we endow the space $X_0$ with the norm defined as (see [@svmountain Lemma 6]) $$\label{norma}
\|g\|_{X_0}=\Big(\int_{{\mathbb R}^n\times {\mathbb R}^n} |g(x)-g(y)|^2K(x-y)dx\,dy\Big)^{1/2},$$ which is obviously related to the so-called *Gagliardo norm* $$\label{gagliardonorm}
\|g\|_{H^s(\Omega)}=\|g\|_{L^2(\Omega)}+
\Big(\int_{\Omega\times
\Omega}\frac{\,\,\,|g(x)-g(y)|^2}{|x-y|^{n+2s}}\,dx\,dy\Big)^{1/2}$$ of the usual fractional Sobolev space $H^s(\Omega)$. For further details on the fractional Sobolev spaces we refer to [@adams; @valpal; @molicalibro] and to the references therein.

Problem  has a variational structure: indeed, it is the Euler-Lagrange equation of the functional $\mathcal J_\lambda:X_0\to {\mathbb R}$ defined as $$\mathcal J_\lambda(u)=\frac 1 2 \int_{{\mathbb R}^n\times {\mathbb R}^n}|u(x)-u(y)|^2 K(x-y)\,dx\,dy-\frac \lambda 2 \int_\Omega |u(x)|^2\,dx-\int_\Omega F(x, u(x))dx\,.$$ Note that when functional $\mathcal J_\lambda$ is Fréchet differentiable at $u\in X_0$, we have that for any $\varphi\in X_0$ $$\begin{aligned}
\langle \mathcal J'_\lambda(u), \varphi\rangle & = \int_{{\mathbb R}^n\times {\mathbb R}^n} \big(u(x)-u(y)\big)\big(\varphi(x)-\varphi(y)\big)K(x-y)\,dx\,dy-\lambda \int_\Omega u(x)\varphi(x)\,dx\\
& \qquad \qquad \qquad \qquad \qquad \qquad\qquad \qquad \qquad \qquad  -\int_\Omega f(x, u(x))\varphi(x)\,dx\,,
\end{aligned}$$ where we have denoted by $\langle \cdot,\cdot\rangle$ the duality between $X_0'$ and $X_0$. Thus, critical points of $\mathcal J_\lambda$ are solutions to problem . In order to find multiplicity of critical points, we will use the $\nabla$-theorem in the form of Theorem \[thmarinosaccon\] (see Section \[sec:proofthmain\]).

Estimates on the nonlinearity
-----------------------------

Here, we recall some estimates on the nonlinear term and its primitive, which will be useful in the sequel. These estimates are quite standard and do not take into account the non-local features of the problem. For a proof we refer to [@svmountain Lemma 3 and Lemma 4].

By assumptions , and we deduce that $$\label{cond22}
\begin{aligned}
& \mbox{for any}\,\, \varepsilon>0\,\, \mbox{there exists}\,\, C_{\varepsilon}>0\,\, \mbox{such that}\\
& \quad |f(x,t)|\leq 2\varepsilon |t|+qC_{\varepsilon} |t|^{q-1}\,\, \mbox{a.e.}\,\, x\in \Omega, t\in {\mathbb R}\end{aligned}$$ and so, as a consequence, $$\label{cond22F}
|F(x,t)|\leq \varepsilon\,|t|^2+C_{\varepsilon}\, |t|^{q}\,,$$ where $F$ is defined as in . This implies that functional $\mathcal J_\lambda$ is Fréchet differentiable at any point $u\in X_0$.

An eigenvalue problem for $-\mathcal L_K$ {#subsec:eigenvalue}
-----------------------------------------

Along the present paper we consider the following eigenvalue problem associated to the integro-differential operator $-\mathcal L_K$: $$\label{problemaautovalori}
\left\{\begin{array}{ll}
-\mathcal L_K u=\lambda\, u & \mbox{in } \Omega\\
u=0 & \mbox{in } {\mathbb R}^n\setminus \Omega\,.
\end{array}\right.$$ We denote by $\big\{ \lambda_k\big\}_{{k\in{\mathbb N}}}$ the sequence of the eigenvalues of the problem , with $$\label{lambdacrescente}
0<\lambda_1<\lambda_2\le \dots \le \lambda_k\le \lambda_{k+1}\le \dots$$ $${\mbox{$\lambda_k\to +\infty$ as $ k\to +\infty\,,$}}$$ and by $e_k$ the eigenfunction corresponding to $\lambda_k$. Moreover, we normalize $\big\{e_k\big\}_{{k\in{\mathbb N}}}$ in such a way that this sequence provides an orthonormal basis of $L^2(\Omega)$ and an orthogonal basis of $X_0$. For a complete study of the spectrum of the integro-differential operator $-\mathcal L_K$ we refer to [@sY Proposition 2.3], [@svlinking Proposition 9 and Appendix A] and [@servadeivaldinociBNLOW Proposition 4]. In particular, we recall that all eigenfunctions are Hölder continuous up to the boundary of $\Omega$.

Finally, we say that eigenvalue $\lambda_k$, $k\geq 2$, has multiplicity $m\in {\mathbb N}$ if $$\lambda_{k-1}<\lambda_k=\dots =\lambda_{k+m-1}<\lambda_{k+m}\,.$$ In this case the set of all the eigenvalues corresponding to $\lambda_k$ agrees with $$\mbox{span}\left\{e_k,\ldots,e_{k+m-1}\right\}\,.$$

In the following, for any $k\in {\mathbb N}$, we set $$\mathbb H_k=\mbox{span}\left\{e_{1},\ldots,e_k\right\}$$ and $$\mathbb P_k=
\left\{u\in X_{0}:\,\,\left\langle u,e_{j}\right\rangle_{X_{0}}=0\,\,\,\,\mbox{for any}\,\, j=1,\ldots,k \right\},$$ where $$\left\langle u,v\right\rangle_{X_{0}}:= \int_{{\mathbb R}^n\times {\mathbb R}^n} \big(u(x)-u(y)\big)\big(\varphi(x)-\varphi(y)\big)K(x-y)\,dx\,dy$$ makes $X_0$ a Hilbert space, see [@svmountain Lemma 7]. In this way, the variational characterization of the eigenvalues (see [@svlinking Proposition 9]) implies that $$\label{poincarevincolata}
\int_{{\mathbb R}^n\times {\mathbb R}^n}|u(x)-u(y)|^2 K(x-y)\,dx\,dy\geq \lambda_{k+1} \int_\Omega |u(x)|^2\,dx \ \mbox{ for all $u\in \mathbb P_k$},$$ while, using the orthogonality properties of the eigenvalues, a standard Fourier decomposition gives $$\label{antipoincare}
\int_{{\mathbb R}^n\times {\mathbb R}^n}|u(x)-u(y)|^2 K(x-y)\,dx\,dy\leq \lambda_k \int_\Omega |u(x)|^2\,dx \ \mbox{ for all $u\in \mathbb H_k$}.$$

Gradient in $X_0$ {#secgrad}
-----------------

Being $X_0$ a Hilbert space and $\mathcal J_\lambda$ of class $C^1$, the gradient $\nabla \mathcal J_\lambda$ of $\mathcal J_\lambda$ is immediately defined as $$\label{gradJ}
\begin{aligned}
\langle \nabla \mathcal J_\lambda(u),v\rangle_{X_0} :&=\langle \mathcal J'_\lambda(u), v\rangle\\
& =\langle u,v\rangle_{X_0}-\lambda \int_\Omega u(x)v(x)\,dx-\int_\Omega f(x,u(x))v(x)\,dx
\end{aligned}$$ for any $u,v\in X_0$.

Let $\nu \in [1, 2^*]$ and $\nu'$ be its conjugate, that is $1/\nu+1/\nu'=1$. Introducing the operator $\mathcal L_K^{-1}:L^{\nu'}(\Omega)\to X_0$, defined as $\mathcal L_K^{-1}g=v$ if and only if $v\in X_0$ solves $$\begin{cases}
\mathcal L_Kv=g(x) & \mbox{ in } \Omega,\\
v=0 & \mbox{ in } {\mathbb R}^n\setminus \Omega,
\end{cases}$$ it is readily seen that $$\label{grad}
\nabla \mathcal J_\lambda(u)=u-\mathcal L_K^{-1}(\lambda u+f(x,u))$$ for all $u\in X_0$. Indeed, setting $w=\mathcal L_K^{-1}(\lambda u+f(x,u))$ and using the definitions of $\mathcal L_K$ and $\mathcal J_\lambda$, for any test function $\varphi\in X_0$ we get that $$\begin{aligned}
\langle w, \varphi \rangle_{X_0} & = \lambda \int_\Omega u(x)\varphi(x)\,dx +\int_\Omega f(x,u(x))\varphi(x)\,dx\\
& = - \langle \mathcal J_\lambda'(u), \varphi\rangle+\langle u, \varphi \rangle_{X_0}\\
& = -\langle \nabla \mathcal J_\lambda(u),v\rangle_{X_0}+\langle u, \varphi \rangle_{X_0}\,,
\end{aligned}$$ that is $$w=-\nabla \mathcal J_\lambda(u)+u\,,$$ which gives the assertion.

Moreover, we note that $\mathcal L_K^{-1}$ is a compact operator for all $\nu\in [1,2^*)$. Indeed, if $\{g_n\}_{n\in {\mathbb N}}$ is bounded in $L^{\nu'}(\Omega)$, standard calculations imply that $\left\{\mathcal L_K^{-1}g_n\right\}_{n\in {\mathbb N}}$ is relatively compact in $X_0$.

For further use, we also note that, again using the definition of $\mathcal L_K^{-1}$, $$\label{scarica}
\langle u, \mathcal L_K^{-1}v\rangle_{X_0}=\int_\Omega u(x)v(x)\,dx$$ for every $u,v\in X_0$.

Compactness condition {#sec:compactness}
=====================

In this section we check the validity of the *Palais–Smale condition* for functional $\mathcal J_\lambda$ at any level, that is we prove that for each $c\in {\mathbb R}$ any Palais–Smale sequence for $\mathcal J_\lambda$ at level $c$ admits a subsequence which is strongly convergent in $X_0$. As usual, we say that $\{u_j\}_{j\in {\mathbb N}}\subset X_0$ is a *Palais–Smale sequence* for $\mathcal J_\lambda$ at level $c\in {\mathbb R}$ if $$\label{Jc0}
\mathcal J_\lambda(u_j)\to c$$ and $$\label{J'00}
\sup\Big\{ \big|\langle\,\mathcal J'_\lambda(u_j),\varphi\,\rangle \big|\,: \;
\varphi\in
X_0\,, \|\varphi\|_{X_0}=1\Big\}\to 0$$ as $j\to +\infty$.

\[lemmaPS\] Let $\lambda>0$ and $f$ be a function satisfying conditions –.

Then, functional $\mathcal J_\lambda$ satisfies the Palais–Smale condition at any level $c\in {\mathbb R}$.

Let $c\in {\mathbb R}$ and let $\{u_j\}_{j\in {\mathbb N}}$ be a Palais–Smale sequence for $\mathcal J_\lambda$ at level $c$. First of all, let us prove that $$\label{ujlimitata}
\mbox{the sequence}\,\,\, \{u_j\}_{j\in {\mathbb N}} \,\,\, \mbox{is bounded in} \,\,\, X_0\,.$$ For this purpose, we note that by and for any $j\in {\mathbb N}$ it easily follows that there exists $\kappa>0$ such that $$\Big|\langle \mathcal J'_\lambda(u_j), \frac{u_j}{\|u_j\|_{X_0}}\rangle\Big| \leq \kappa\,,$$ and $$|\mathcal J_\lambda(u_j)|\leq \kappa\,,$$ so that, as a consequence of these two relations, we also have $$\label{kappaPS}
\mathcal J_\lambda(u_j)-\frac 1 \mu \langle \mathcal J'_\lambda(u_j), u_j\rangle\leq \kappa \left(1+ \|u_j\|_{X_0}\right)\,,$$ where $\mu$ is a parameter such that $\mu\in (2, q)$.

Now, thanks to and we get $$\label{jj'0L}
\begin{aligned}
\mathcal J_\lambda(u_j)-\frac 1 \mu \langle \mathcal J_\lambda'(u_j), u_j\rangle & = \left(\frac 1 2 -\frac 1 \mu\right)\Big(\|u_j\|_{X_0}^2-\lambda \|u_j\|_{L^2(\Omega)}^2\Big)\\
& \qquad  -\int_\Omega \Big(F(x, u_j(x))-\frac 1 \mu\, f(x, u_j(x)) \,u_j(x)\Big)\,dx\\
& \geq \left(\frac 1 2 -\frac 1 \mu\right)\Big(\|u_j\|_{X_0}^2-\lambda \|u_j\|_{L^2(\Omega)}^2\Big)\\
& \qquad +\left(\frac q \mu-1\right)\int_\Omega F(x, u_j(x))\,dx\\
& \geq \left(\frac 1 2 -\frac 1 \mu\right)\Big(\|u_j\|_{X_0}^2-\lambda \|u_j\|_{L^2(\Omega)}^2\Big)\\
& \qquad +a_3\left(\frac q \mu-1\right)\|u_j\|_{L^q(\Omega)}^q-a_4\left(\frac q \mu-1\right)\,|\Omega|\,.
\end{aligned}$$ Note that for any $\varepsilon>0$ the Young inequality gives $$\label{youngPS}
\|u_j\|_{L^2(\Omega)}^2\leq \frac{2\varepsilon}{q}\,\|u_j\|_{L^q(\Omega)}^q+\frac{q-2}{q}\,\varepsilon^{-2/(q-2)}\,|\Omega|\,,$$ so that, by and , we obtain that $$\label{young2}
\begin{aligned}
\mathcal J_\lambda(u_j)-\frac 1 \mu \langle \mathcal J_\lambda'(u_j), u_j\rangle &
\geq \left(\frac 1 2 -\frac 1 \mu\right)\|u_j\|_{X_0}^2-\lambda\left(\frac 1 2 -\frac 1 \mu\right) \frac{2\varepsilon}{q}\,\|u_j\|_{L^q(\Omega)}^q\\
& \qquad -\lambda\left(\frac 1 2 -\frac 1 \mu\right) \frac{q-2}{q}\,\varepsilon^{-2/(q-2)}\,|\Omega|\\
& \qquad +a_3\left(\frac q \mu-1\right)\|u_j\|_{L^q(\Omega)}^q-a_4\left(\frac q \mu-1\right)\,|\Omega|\\
& = \left(\frac 1 2 -\frac 1 \mu\right)\|u_j\|_{X_0}^2\\
& \qquad +\Big[a_3\left(\frac q \mu-1\right)-\lambda\left(\frac 1 2 -\frac 1 \mu\right)\frac{2\varepsilon}{q}\Big]\|u_j\|_{L^q(\Omega)}^q-C_\varepsilon\,,
\end{aligned}$$ where $C_\varepsilon$ is a constant such that $C_\varepsilon\to +\infty$ as $\varepsilon\to 0$, being $q>\mu>2$.

Now, choosing $\varepsilon$ so small that $$a_3\left(\frac q \mu-1\right)-\lambda\left(\frac 1 2 -\frac 1 \mu\right)\frac{2\varepsilon}{q}>0\,,$$ by we get $$\label{jj'20}
\mathcal J_\lambda(u_j)-\frac 1 \mu \langle \mathcal J_\lambda'(u_j), u_j\rangle \geq \left(\frac 1 2 -\frac 1 \mu\right)\|u_j\|_{X_0}^2-C_\varepsilon\,.$$

Finally, by and for any $j\in {\mathbb N}$ $$\|u_j\|_{X_0}^2 \leq \kappa_*\left(1+\|u_j\|_{X_0}\right)$$ for a suitable positive constant $\kappa_*$. Hence, assertion  is proved.

Now, let us finish the proof of the Palais–Smale condition for $\mathcal J_\lambda$. Since $\{u_j\}_{j\in {\mathbb N}}$ is bounded in $X_0$ and $X_0$ is a reflexive space, up to a subsequence, still denoted by $u_j$, there exists $u_\infty \in X_0$ such that $$\label{convergenze0}
\begin{aligned}
 & \int_{{\mathbb R}^n\times {\mathbb R}^n}\big(u_j(x)-u_j(y)\big)\big(\varphi(x)-\varphi(y)\big) K(x-y)\,dx\,dy \to \\
& \qquad \qquad
\int_{{\mathbb R}^n\times {\mathbb R}^n}\big(u_\infty(x)-u_\infty(y)\big)\big(\varphi(x)-\varphi(y)\big)
K(x-y)\,dx\,dy  \quad \mbox{for any}\,\, \varphi\in X_0\,,
\end{aligned}$$ while, by [@svmountain Lemma 8], up to a subsequence, $$\label{convergenze0bis}
\begin{aligned}
& u_j \to u_\infty \quad \mbox{in}\,\, L^\nu({\mathbb R}^n)\quad \mbox{for any}\,\,\nu\in [1, 2^*) \\
& u_j \to u_\infty \quad \mbox{a.e. in}\,\, {\mathbb R}^n
\end{aligned}$$ as $j\to +\infty$. Finally, by [@brezis Theorem IV.9] we know that for any $\nu\in [1, 2^*)$ there exists $\ell_\nu\in L^\nu({\mathbb R}^n)$ such that $$\label{dominata20}
|u_j(x)|\leq \ell_\nu(x) \quad \mbox{a.e. in}\,\, {\mathbb R}^n\,\quad \mbox{for any}\,\,j\in {\mathbb N}\,.$$

By , , , the fact that the map $t\mapsto f(\cdot, t)$ is continuous in $t\in {\mathbb R}$ (see ) and the Dominated Convergence Theorem, we get $$\label{convf0}
\int_\Omega f(x, u_j(x))u_j(x)\,dx \to \int_\Omega f(x, u_\infty(x))u_\infty(x)\,dx$$ and $$\label{convfu0}
\int_\Omega f(x, u_j(x))u_\infty(x)\,dx \to \int_\Omega f(x, u_\infty(x))u_\infty(x)\,dx$$ as $j\to +\infty$. Furthermore, by and we have that $$\begin{aligned}
 0\leftarrow \langle \mathcal J'_\lambda(u_j), u_j\rangle & = \int_{{\mathbb R}^n\times {\mathbb R}^n}|u_j(x)-u_j(y)|^2 K(x-y)\,dx\,dy -\lambda \int_\Omega |u_j(x)|^2\,dx\\
& \qquad \qquad \qquad \qquad \qquad \qquad \qquad - \int_\Omega f(x, u_j(x))u_j(x)\,dx
\end{aligned}$$ so that, by and we deduce that $$\label{convnorma10}
\int_{{\mathbb R}^n\times {\mathbb R}^n}|u_j(x)-u_j(y)|^2 K(x-y)\,dx\,dy\to \lambda \int_\Omega |u_\infty(x)|^2\,dx+\int_\Omega f(x, u_\infty(x))u_\infty(x)\,dx$$ as $j\to +\infty$, while, by , (both with test function $\varphi=u_\infty$), and , we get $$\label{convnorma20}
\begin{aligned}
\int_{{\mathbb R}^n\times {\mathbb R}^n}|u_\infty(x)-u_\infty(y)|^2 K(x-y)\,dx\,dy & = \lambda \int_\Omega |u_\infty(x)|^2\,dx\\
& \qquad \qquad +\int_\Omega f(x, u_\infty(x))u_\infty(x)\,dx\,.
\end{aligned}$$ Hence, and give that $$\label{convnormaX00}
\|u_j\|_{X_0}\to \|u_\infty\|_{X_0}$$ as $j\to +\infty$. With this, it is easy to see that $$\begin{aligned}
\|u_j-u_\infty\|_{X_0}^2 & = \|u_j\|_{X_0}^2 + \|u_\infty\|_{X_0}^2 -2 \int_{{\mathbb R}^n\times {\mathbb R}^n} \big(u_j(x)-u_j(y)\big)\big(u_\infty(x)-u_\infty(y)\big) K(x-y)\,dx\,dy\\
& \to 2\|u_\infty\|_{X_0}^2-2\int_{{\mathbb R}^n\times {\mathbb R}^n}|u_\infty(x)-u_\infty(y)|^2 K(x-y)\,dx\,dy=0
\end{aligned}$$ as $j\to +\infty$, thanks to and . Then, the proof of Proposition \[lemmaPS\] is complete.

Geometry of the $\nabla$-theorem {#sec:geometry}
================================

In this section we check that functional $\mathcal J_\lambda$ has the geometric structure required by the $\nabla$-theorem stated in Theorem \[thmarinosaccon\] (see Section \[sec:proofthmain\]). Precisely, we want to show that if there exist $k$ and $m$ in ${\mathbb N}$ such that $$\lambda_{k-1}<\lambda<\lambda_k=\dots = \lambda_{k+m-1}<\lambda_{k+m}$$ and $\lambda$ is sufficiently close to $\lambda_k$, then functional $\mathcal J_\lambda$ agrees with the geometric framework of Theorem \[thmarinosaccon\], taking $$\begin{aligned}
& X_1:=\mathbb H_{k-1}\\
& X_2:=\mbox{span}\left\{e_k,\ldots,e_{k+m-1}\right\}\\
& X_3:=\mathbb P_{k+m-1}\,.
\end{aligned}$$

\[propgeometria\] Let $k$ and $m$ in ${\mathbb N}$ be such that $\lambda_{k-1}<\lambda<\lambda_k=\dots = \lambda_{k+m-1}<\lambda_{k+m}$ and let $f$ be a function satisfying conditions –.

Then, there exist $\rho, R$, with $R>\rho>0$, such that $${\displaystyle \sup_{\{u\in X_1, \|u\|_{X_0}\leq R\} \cup \{u\in X_1\oplus X_2 : \|u\|=R\}}} \mathcal J_\lambda(u)<{\displaystyle \inf_{\{u\in X_2\oplus X_3 : \|u\|_{X_0}=\rho\}}\mathcal J_\lambda(u)}\,.$$

First of all, let us show that $$\label{infpositivo}
{\displaystyle \inf_{\{u\in X_2\oplus X_3 : \|u\|_{X_0}=\rho\}}\mathcal J_\lambda(u)}>0\,.$$ For this purpose, let $u$ be a function in $X_2 \oplus X_3=\mathbb P_{k-1}$. By , we get that for any $\varepsilon>0$ $$\label{infJfrac0}
\begin{aligned}
\mathcal J_\lambda(u)& \geq \frac 1 2\int_{{\mathbb R}^n\times {\mathbb R}^n}|u(x)-u(y)|^2 K(x-y)dx\,dy-\frac \lambda 2\int_\Omega |u(x)|^2\,dx\\
& \qquad \qquad \qquad \qquad -\varepsilon\int_\Omega |u(x)|^2dx - C_{\varepsilon}\int_\Omega |u(x)|^q\,dx\,.
\end{aligned}$$ Moreover, by , we get that for any $u\in \mathbb P_{k-1}$ $$\lambda_k\,\int_\Omega |u(x)|^2\,dx \leq \int_{{\mathbb R}^n\times {\mathbb R}^n}|u(x)-u(y)|^2 K(x-y)dx\,dy\,,$$ so that this inequality and give $$\label{infJfrac0bis}
\begin{aligned}
\mathcal J_\lambda(u)& \geq \frac 1 2\left(1-\frac{\lambda}{\lambda_k}\right)\|u\|_{X_0}^2-\varepsilon\|u\|_{L^2(\Omega)}^2
 -C_{\varepsilon}\|u\|_{L^q(\Omega)}^q \\
& \geq \frac 1 2\left(1-\frac{\lambda}{\lambda_k}\right)\|u\|_{X_0}^2-\varepsilon|\Omega|^{(2^*-2)/2^*}
\|u\|_{L^{2^*}(\Omega)}^2 -|\Omega|^{(2^*-q)/2^*}C_{\varepsilon}\|u\|_{L^{2^*}(\Omega)}^q\,.
\end{aligned}$$ Here we used also the fact that $L^{2^*}(\Omega)\hookrightarrow L^\nu(\Omega)$ continuously, being $\Omega$ bounded and $\nu\in [2, 2^*)$ (here $\nu$ takes the values $2$ and $q$).

Using , and [@svmountain Lemma 6], we obtain that for any $\varepsilon>0$, $$\label{infJfrac20}
\begin{aligned}
\mathcal J_\lambda(u) & \geq \frac 1 2\left(1-\frac{\lambda}{\lambda_k}\right)\|u\|_{X_0}^2-\varepsilon c|\Omega|^{(2^*-2)/2^*}
 \int_{{\mathbb R}^n \times {\mathbb R}^n}\frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}}\, dx\,dy \\
& \qquad -C_{\varepsilon}c^{q/2}|\Omega|^{(2^*-q)/2^*} \left( \int_{{\mathbb R}^n \times {\mathbb R}^n}\frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}}\, dx\,dy\right)^{q/2}\\
& \ge \left[\frac 1 2\left(1-\frac{\lambda}{\lambda_k}\right) -\frac{\varepsilon
c|\Omega|^{(2^*-2)/2^*}}{\theta}\right]\int_{{\mathbb R}^n\times {\mathbb R}^n} |u(x)-u(y)|^2 K(x-y)\,dx\,dy\\
& \qquad -\frac{C_{\varepsilon}c^{q/2}|\Omega|^{(2^*-q)/2^*}}{\theta} \left( \int_{{\mathbb R}^n\times {\mathbb R}^n} |u(x)-u(y)|^2 K(x-y)\, dx\,dy\right)^{q/2}\,,
\end{aligned}$$ where $c$ is a suitable universal positive constant.

Choosing $\varepsilon>0$ such that $$2\varepsilon c
|\Omega|^{(2^*-2)/2^*}<\theta\left(1-\frac{\lambda}{\lambda_k}\right),$$ inequality  reads as $$\mathcal J_\lambda(u) \geq \alpha \|u\|_{X_0}^2\left(1-\kappa
\|u\|_{X_0}^{q-2}\right)\,,$$ for suitable positive constants $\alpha$ and $\kappa$.

Now, let $\rho>0$ be sufficiently small, i.e. $\rho$ such that $1-\kappa\rho^{q-2}>0$. Then, for any $u\in \mathbb P_{k-1}$ such that $\|u\|_{X_0}=\rho$ we get that $$\mathcal J_\lambda(u)\geq \alpha \rho^2(1-\kappa \rho^{q-2})>0\,,$$ so that is proved.

Now, let us show that $$\label{supnegativo}
{\displaystyle \sup_{\{u\in X_1, \|u\|_{X_0}\leq R\} \cup \{u\in X_1\oplus X_2 : \|u\|=R\}}} \mathcal J_\lambda(u)\leq 0\,.$$ First of all, let us take $u\in \mathbb H_{k-1}$. Then $$u(x)=\sum_{i=1}^{k-1}u_i e_i(x)\,,$$ with $u_i\in {\mathbb R}$, $i=1, \dots,
k-1$ and so, by and , we deduce that $$\label{J<01}
\begin{aligned}
\mathcal J_\lambda(u) & \leq \frac{\lambda_{k-1}-\lambda}{2}\int_\Omega |u(x)|^2\,dx \leq 0\,,
\end{aligned}$$ since $\lambda_{k-1}< \lambda$.

Finally, let us consider $u\in X_1\oplus X_2=\mathbb H_k$. By we have $$\mathcal J_\lambda(u)  \leq \frac 1 2\|u\|_{X_0}^2-a_3\|u\|_{L^q(\Omega)}^q+a_4|\Omega|,$$ and the claim follows recalling that $q>2$ and that $\mathbb H_k$ is a finite-dimensional subspace of $X_0$. This and give .

Then, the assertion of Proposition \[propgeometria\] comes trivially from and .

$\nabla$-condition {#sec:nablacondition}
==================

One of the main ingredient of the $\nabla$-theorem (see [@marsac:sns Theorem 2.10]) we employ in order to get our multiplicity result is the so-called $\nabla$-condition introduced in [@marsac:sns Definition 2.4]. This section is devoted to the verification of this condition for functional $\mathcal J_\lambda$. For this purpose, in the sequel we denote by $$P_C:X_0\to C$$ the orthogonal projection of $X_0$ onto $C$.

Let $C$ be a closed subspace of $X_0$ and $a,b\in {\mathbb R}\cup\{-\infty, +\infty\}$. We say that functional $\mathcal J_\lambda$ verifies condition $(\nabla)(\mathcal J_\lambda, C, a, b)$ if there exists $\gamma>0$ such that $$\inf\Big\{\|P_C \nabla \mathcal J_\lambda (u)\|_{X_0} : a\leq \mathcal J_\lambda(u)\leq b,\,\, dist (u, C)\leq \gamma\Big\}>0\,.$$

Roughly speaking, the condition $(\nabla)(\mathcal J_\lambda, C, a, b)$ requires that $\mathcal J_\lambda$ has no critical points $u\in C$ such that $a\leq \mathcal J_\lambda(u)\leq b$, with some uniformity. In order to prove this condition for $\mathcal J_\lambda$, we need two preliminary lemmas.

\[lemmanabla1\] Let $k$ and $m$ in ${\mathbb N}$ be such that $\lambda_{k-1}<\lambda_k=\dots = \lambda_{k+m-1}<\lambda_{k+m}$ and let $f$ be a function satisfying conditions –.

Then, for any $\sigma>0$ there exists $\varepsilon_\sigma>0$ such that for any $\lambda\in [\lambda_{k-1}+\sigma, \lambda_{k+m}-\sigma]$ the unique critical point $u$ of $\mathcal J_\lambda$ constrained on $\mathbb H_{k-1}\oplus \mathbb P_{k+m-1}$ and with $\mathcal J_\lambda(u)\in [-\varepsilon_\sigma, \varepsilon_\sigma]$, is the trivial one.

We argue by contradiction and we suppose that there exists $\bar \sigma>0$, a sequence $\{\mu_j\}_{j\in {\mathbb N}}$ in ${\mathbb R}$ with $$\label{lambdaj}
\mu_j\in [\lambda_{k-1}+\bar \sigma, \lambda_{k+m}-\bar \sigma]$$ and a sequence $\{u_j\}_{j\in {\mathbb N}}$ such that $$\label{ujoplus}
u_j\in \mathbb H_{k-1}\oplus \mathbb P_{k+m-1}\setminus\{0\}\,,$$ $$\label{ujvincolato}
\langle\,\mathcal J'_{\mu_j}(u_j),\varphi\,\rangle =0\qquad \mbox{for any}\,\, \varphi\in \mathbb H_{k-1}\oplus \mathbb P_{k+m-1}\,,$$ for any $j\in {\mathbb N}$, and $$\label{ujvalorecritico}
\begin{aligned}
\mathcal J_{\mu_j}(u_j) & =\frac 1 2 \int_{{\mathbb R}^n\times {\mathbb R}^n}|u_j(x)-u_j(y)|^2 K(x-y)\,dx\,dy -\frac{\mu_j}{2} \int_\Omega |u_j(x)|^2\,dx\\
& \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad- \int_\Omega F(x, u_j(x))\,dx\to 0
\end{aligned}$$ as $j\to +\infty$.

Taking $\varphi=u_j$ in (this is possible thanks to ) and using , we get that for any $j\in {\mathbb N}$ $$\begin{aligned}
0 & =\int_{{\mathbb R}^n\times {\mathbb R}^n}|u_j(x)-u_j(y)|^2 K(x-y)\,dx\,dy -\mu_j \int_\Omega |u_j(x)|^2\,dx- \int_\Omega f(x, u_j(x))u_j(x)\,dx\\
& = 2\mathcal J_{\mu_j}(u_j)+\int_\Omega \Big(2F(x, u_j(x))-f(x, u_j(x))u_j(x)\Big)\,dx\\
& \leq 2\mathcal J_{\mu_j}(u_j)+(2-q)\int_\Omega F(x, u_j(x))\,dx\,.
\end{aligned}$$ Hence, by this inequality, the fact that $q>2$ and again , we deduce that $$0<(q-2)\int_\Omega F(x, u_j(x))\,dx\leq 2\mathcal J_{\mu_j}(u_j)\to 0$$ thanks to , so that we get $$\label{Fujlimite}
\int_\Omega F(x, u_j(x))\,dx\to 0$$ as $j\to +\infty$.

Now, since holds true, for any $j\in{\mathbb N}$ there exist $v_j\in \mathbb H_{k-1}$ and $w_j\in\mathbb P_{k+m-1}$ such that $$u_j=v_j+w_j\,.$$ Letting $\varphi=v_j-w_j$ in and taking into account the orthogonality properties of $v_j$ and $w_j$, we have that for any $j\in {\mathbb N}$ $$\label{split}
\begin{aligned}
0 & =\langle\mathcal J'_{\mu_j}(u_j), v_j-w_j\rangle\\
& = \int_{{\mathbb R}^n\times {\mathbb R}^n}|v_j(x)-v_j(y)|^2 K(x-y)\,dx\,dy- \int_{{\mathbb R}^n\times {\mathbb R}^n}|w_j(x)-w_j(y)|^2 K(x-y)\,dx\,dy\\
& \qquad \qquad -\mu_j \int_\Omega |v_j(x)|^2\,dx + \mu_j \int_\Omega |w_j(x)|^2\,dx- \int_\Omega f(x, u_j(x))(v_j(x)-w_j(x))\,dx\,.
\end{aligned}$$ By and , gives that for any $j\in {\mathbb N}$ $$\label{carvar2}
\begin{aligned}
\int_\Omega f(x, u_j(x))(v_j(x)-w_j(x))\,dx & = \int_{{\mathbb R}^n\times {\mathbb R}^n}|v_j(x)-v_j(y)|^2 K(x-y)\,dx\,dy\\
& \qquad \qquad - \int_{{\mathbb R}^n\times {\mathbb R}^n}|w_j(x)-w_j(y)|^2 K(x-y)\,dx\,dy\\
& \qquad \qquad -\mu_j \int_\Omega |v_j(x)|^2\,dx + \mu_j \int_\Omega |w_j(x)|^2\,dx\\
& \leq \frac{\lambda_{k-1}-\mu_j}{\lambda_{k-1}}\|v_j\|_{X_0}^2+ \frac{\mu_j-\lambda_{k+m}}{\lambda_{k+m}}\|w_j\|_{X_0}^2\\
& \leq -\frac{\bar \sigma}{\lambda_{k-1}}\|v_j\|_{X_0}^2- \frac{\bar \sigma}{\lambda_{k+m}}\|w_j\|_{X_0}^2\\
&\leq -\frac{\bar \sigma}{\lambda_{k+m}}\|u_j\|_{X_0}^2\,,
\end{aligned}$$ thanks again to the properties of the projections $v_j$ and $w_j$ of $u_j$, respectively on $\mathbb H_{k-1}$ and on $\mathbb P_{k+m-1}$, and to .

On the other hand, by the Hölder inequality, and the fact that $X_0\hookrightarrow L^q(\Omega)$ compactly, there exists a suitable positive constant $\tilde\kappa$, independent of $j$, such that $$\begin{aligned}
\Big|\int_\Omega f(x, u_j(x))(v_j(x)-w_j(x))\,dx\Big| & \leq
\|f(\cdot, u_j(\cdot))\|_{L^{q/(q-1)}(\Omega)}\, \|v_j-w_j\|_{L^q(\Omega)}\\
& \leq  \tilde\kappa \|f(\cdot, u_j(\cdot))\|_{L^{q/(q-1)}(\Omega)}\, \|v_j-w_j\|_{X_0}\\
& = \tilde\kappa \|f(\cdot, u_j(\cdot))\|_{L^{q/(q-1)}(\Omega)}\, \|u_j\|_{X_0}\,,\\
\end{aligned}$$ so that, by this and , we deduce that $$\tilde\kappa \|f(\cdot, u_j(\cdot))\|_{L^{q/(q-1)}(\Omega)}\, \|u_j\|_{X_0}\geq \frac{\bar \sigma}{\lambda_{k+m}}\|u_j\|_{X_0}^2.$$ Being $u_j\not \equiv 0$ by assumption (see ), we get $$\label{f>}
\|f(\cdot, u_j(\cdot))\|_{L^{q/(q-1)}(\Omega)}\geq \frac{\bar \sigma}{\tilde\kappa \lambda_{k+m}}\|u_j\|_{X_0}$$ for any $j\in {\mathbb N}$.

With the previous estimates, we are now ready to show that $$\label{ujboundednabla}
\mbox{the sequence}\,\, \{\|u_j\|_{X_0}\}_{j\in{\mathbb N}}\,\, \mbox{is bounded in}\,\, {\mathbb R}.$$ For this it is enough to use and , which yield for any $j\in {\mathbb N}$ $$\label{fujnabla}
\begin{aligned}
\int_\Omega |f(x, u_j(x))|^{q/(q-1)}\,dx & \leq \int_\Omega \left(a_1+a_2|u_j(x)|^{q-1}\right)^{q/(q-1)} \\
& \leq \tilde a_1 +\tilde a_2 \int_\Omega |u_j(x)|^q\,dx\\
& \leq \tilde a_3 + \tilde a_4 \int_\Omega F(x,u_j(x))\,dx\,,
\end{aligned}$$ for suitable positive constants $\tilde a_i$, $i=1,\ldots, 4$. By , and we get assertion .

In view of and , we can assume that there exists $u_\infty\in \mathbb H_{k-1}\oplus \mathbb P_{k+m-1}$ such that $$\label{convergenze0nabla}
\begin{aligned}
 & \int_{{\mathbb R}^n\times {\mathbb R}^n}\big(u_j(x)-u_j(y)\big)\big(\varphi(x)-\varphi(y)\big) K(x-y)\,dx\,dy \to \\
& \qquad \qquad
\int_{{\mathbb R}^n\times {\mathbb R}^n}\big(u_\infty(x)-u_\infty(y)\big)\big(\varphi(x)-\varphi(y)\big)
K(x-y)\,dx\,dy  \quad \mbox{for any}\,\, \varphi\in X_0\,,
\end{aligned}$$ while, by [@svmountain Lemma 8] and [@brezis Theorem IV.9], up to a subsequence, $$\label{convergenze0bisnabla}
\begin{aligned}
& u_j \to u_\infty \quad \mbox{in}\,\, L^q({\mathbb R}^n)\\
& u_j \to u_\infty \quad \mbox{a.e. in}\,\, {\mathbb R}^n
\end{aligned}$$ as $j\to +\infty$ and there exists $\ell\in L^q({\mathbb R}^n)$ such that $$\label{dominata20nabla}
|u_j(x)|\leq \ell(x) \quad \mbox{a.e. in}\,\, {\mathbb R}^n\,\quad \mbox{for any}\,\,j\in {\mathbb N}\,.$$

Moreover, by and we get that for any $\varepsilon>0$ there exists $C_\varepsilon$ such that $$\label{assurdo1}
\begin{aligned}
0<\frac{\bar \sigma}{\tilde \kappa \lambda_{k+m}} & \leq \frac{\|f(\cdot, u_j(\cdot))\|_{L^{q/(q-1)}(\Omega)}}{\|u_j\|_{X_0}}\\
& \leq \frac{{\displaystyle \left(\int_\Omega \left(2\varepsilon |u_j(x)|+qC_{\varepsilon} |u_j(x)|^{q-1}\right)^{q/(q-1)}\,dx\right)^{(q-1)/q}}}{\|u_j\|_{X_0}}\\
& \leq \frac{\left( 2^{1/(q-1)}\left((2\varepsilon)^{q/(q-1)}\|u_j\|_{L^{q/(q-1)}(\Omega)}^{q/(q-1)}+
(qC_{\varepsilon})^{q/(q-1)}\|u_j\|_{L^q(\Omega)}^q\right)\right)^{(q-1)/q}}{\|u_j\|_{X_0}}\\
& \leq \frac{2\varepsilon\|u_j\|_{L^{q/(q-1)}(\Omega)}+
qC_{\varepsilon}\|u_j\|_{L^q(\Omega)}^{q-1}}{\|u_j\|_{X_0}}\\
& \leq C\left(2\varepsilon+
qC_{\varepsilon}\|u_j\|_{X_0}^{q-2}\right)\,,
\end{aligned}$$ thanks to the continuous embedding $X_0\hookrightarrow L^\nu(\Omega)$ for any $\nu\in [1, 2^*)$, and for some universal positive constant $C$.

By , , , and the Dominated Convergence Theorem, it is easily seen that $$\label{Fdominata}
\int_\Omega F(x, u_j(x))\,dx \to \int_\Omega F(x, u_\infty(x))\,dx$$ and $$\label{fdominata}
\int_\Omega |f(x, u_j(x))|^{q/(q-1)}\,dx \to \int_\Omega |f(x, u_\infty(x))|^{q/(q-1)}\,dx$$ as $j\to +\infty$. Relation , combined with , the fact that $F(x,0)=0$ a.e. $x\in \Omega$, and , yields that $$\label{uinftynabla}
u_\infty \equiv 0\,.$$

Now, two cases can occur. If $$\label{uinfty0}
u_j \to u_\infty\equiv 0\qquad \mbox{strongly in}\,\, X_0$$ as $j\to +\infty$, then, by we get that $$0<\frac{\bar \sigma}{\tilde\kappa \lambda_{k+m}} \leq 2C\varepsilon\,,$$ which gives a contradiction, due to the fact that $\varepsilon$ is arbitrary. Otherwise, there exists $\eta>0$ such that $\|u_j\|_{X_0}\geq \eta$ for $j$ large enough. Then, by this, , , and the fact that $f(x,0)=0$ a.e. $x\in \Omega$ (by ), we get that $$\frac{\bar \sigma\eta}{\tilde\kappa \lambda_{k+m}}\leq 0\,,$$ which is a contradiction. This completes the proof of Lemma \[lemmanabla1\].

The second lemma we need in order to prove the $\nabla$-condition is the following one:

\[lemmanabla2\] Let $k$ and $m$ in ${\mathbb N}$ be such that $\lambda_{k-1}<\lambda_k=\dots = \lambda_{k+m-1}<\lambda_{k+m}$, let $\lambda \in {\mathbb R}$ and $f$ be a function satisfying conditions –. Moreover, let $\{u_j\}_{j\in {\mathbb N}}$ be a sequence in $X_0$ such that $$\label{condlemma1}
\{\mathcal J_\lambda(u_j)\}_{j\in {\mathbb N}}\,\,\, \mbox{is bounded in}\,\,\, {\mathbb R}\,,$$ $$\label{condlemma2}
P_{\footnotesize{\rm{span}}\left\{e_k,\ldots,\,e_{k+m-1}\right\}} u_j\to 0\,\,\, \mbox{in}\,\,\, X_0$$ and $$\label{condlemma3}
P_{\mathbb H_{k-1}\oplus \mathbb P_{k+m-1}}\nabla \mathcal J_\lambda(u_j)\to 0\,\,\, \mbox{in}\,\,\, X_0$$ as $j\to +\infty$.

Then, $\{u_j\}_{j\in {\mathbb N}}$ is bounded in $X_0$.

Assume by contradiction that $\{u_j\}_{j\in {\mathbb N}}$ is unbounded in $X_0$; without loss of generality, we can assume that $$\label{ujtoinfty}
\|u_j\|_{X_0}\to +\infty$$ as $j\to +\infty$ and that there exists $u_\infty\in X_0$ such that $$\label{convnorm}
\begin{aligned}
& \frac{u_j}{\|u_j\|_{X_0}}\rightharpoonup u_\infty\,\,\, \mbox{ in }X_0\\
& \frac{u_j}{\|u_j\|_{X_0}}\to u_\infty\,\,\, \mbox{ in }L^\nu(\Omega) \mbox{ for any $\nu\in[1,2^*)$}
\end{aligned}$$ as $j\to +\infty$ and for any $\nu\in[1,2^*)$ there exists $\ell_\nu\in L^\nu({\mathbb R}^n)$ such that $$\label{dominata20nablaadd}
|u_j(x)|\leq \ell_\nu(x) \quad \mbox{a.e. in}\,\, {\mathbb R}^n\,\quad \mbox{for any}\,\,j\in {\mathbb N}\,.$$

Now, for simplicity, we set $P_{\footnotesize{\mbox{span}}\left\{e_k,\ldots,\,e_{k+m-1}\right\}}=:P$ and $P_{\mathbb H_{k-1}\oplus \mathbb P_{k+m-1}}=:Q$, and write $$u_j=Pu_j+Qu_j\,,$$ where $Pu_j\to 0$ as $j\to\infty$ (see ). Recalling and , we have $$\label{add11}
\begin{aligned}
\langle Q\nabla \mathcal J_\lambda(u_j),u_j\rangle_{X_0}&= \langle \nabla
\mathcal J_\lambda(u_j),u_j\rangle_{X_0} - \langle P\nabla
\mathcal J_\lambda(u_j),u_j\rangle_{X_0}\\
&=\|u_j\|_{X_0}^2-\lambda \int_\Omega |u_j(x)|^2\,dx
-\int_\Omega f(x,u_j(x))u_j(x)\, dx \\
&-\langle P\big(u_j-\mathcal L_K^{-1}(\lambda
 u_j+f(x,u_j))\big), u_j\rangle_{X_0}.
\end{aligned}$$ Since $\langle Pu,v\rangle_{X_0}=\langle u,Pv\rangle_{X_0}$ for any $u,v\in X_0$, we have $$\label{add22}
\begin{aligned}
\langle P\big(u_j-\mathcal L_K^{-1}(\lambda
u_j+f(x,u_j))\big), u_j\rangle_{X_0} & =\|Pu_j\|^2_{X_0}-\lambda \langle Pu_j, \mathcal L_K^{-1}u_j\rangle_{X_0}\\
& \qquad -\langle Pu_j, \mathcal L_K^{-1}f(x,u_j)\rangle_{X_0},
\end{aligned}$$ while, by , we obtain $$\label{add33}
\begin{aligned}
\lambda \langle Pu_j, \mathcal L_K^{-1}u_j\rangle_{X_0}+\langle Pu_j, \mathcal L_K^{-1}f(x,u_j)\rangle_{X_0} & =\lambda \int_\Omega |Pu_j(x)|^2\,dx\\
& \qquad +\int_\Omega f(x,u_j(x))Pu_j(x)\,dx.
\end{aligned}$$ Therefore, by -, we get $$\label{sec}
\begin{aligned}
\langle Q\nabla \mathcal J_\lambda(u_j),u_j\rangle_{X_0}& =2\mathcal J_\lambda(u_j)+
2\int_\Omega F(x,u_j(x))\, dx -\int_\Omega f(x,u_j(x))u_j(x)\,dx\\
& \qquad -\|Pu_j\|_{X_0}^2 +\lambda \int_\Omega |Pu_j(x)|^2\, dx+\int_\Omega f(x,u_j(x))Pu_j(x)\,dx.
\end{aligned}$$

By – and we easily get that $$\label{Ffujto0}
\frac{\displaystyle{2\int_\Omega F(x,u_j(x))\, dx -\int_\Omega f(x,u_j(x))u_j(x)\,dx+\int_\Omega f(x,u_j(x))Pu_j(x)\,dx}}{\|u_j\|_{X_0}^q} \to 0$$ as $j\to +\infty$.

Now, let us show that $$\label{uinfty0}
u_\infty\equiv 0\,.$$ For this purpose, we firstly claim that $$\label{add44}
\frac{\displaystyle \int_\Omega
f(x,u_j(x))Pu_j(x)\,dx}{\|u_j\|_{X_0}^q} \to 0$$ as $j\to +\infty$. Indeed, by and , we have that a.e. $x\in \Omega$ $$\begin{aligned}
|f(x,u_j(x))Pu_j(x)| & \leq \|Pu_j\|_\infty\big(a_1+a_2|u_j(x)|^{q-1}\big)\\
&\leq \|Pu_j\|_\infty\big(a_1+a_2|\ell_q(x)|^{q-1}\big)\,,
\end{aligned}$$ while, by and the fact that all norms in $\mathbb H_{k+m-1}$ are equivalent, $$\|Pu_j\|_\infty\to 0$$ as $j\to +\infty$. Note that $Pu_j\in L^\infty(\Omega)$, since all eigenfunctions of $\mathcal L_K$ are bounded (see [@sY Proposition 2.4]). Hence, holds.

By this and we obtain that $$0\leftarrow \frac{\displaystyle{2\int_\Omega F(x,u_j(x))\, dx -\int_\Omega f(x,u_j(x))u_j(x)\,dx}}{\|u_j\|_{X_0}^q} \leq \frac{\displaystyle{(2-q)\int_\Omega F(x,u_j(x))\, dx}}{\|u_j\|_{X_0}^q}\leq 0\,,$$ as $j\to +\infty$, also thanks to . Hence, $$\frac{\displaystyle \int_\Omega F(x,u_j(x))\,
dx}{\|u_j\|_{X_0}^q}\to 0$$ as $j\to +\infty$.

As a consequence of this, and we have that $$\frac{\displaystyle \int_\Omega |u_j(x)|^q\,
dx}{\|u_j\|_{X_0}^q}\to 0,$$ as $j\to +\infty$, which yields , thanks to .

Now, by and , we get $$\frac{\mathcal J_\lambda(u_j)}{\|u_j\|_{X_0}^2}=\frac 1 2-\frac \lambda 2\frac{\displaystyle \int_\Omega |u_j(x)|^2\,
dx}{\|u_j\|_{X_0}^2}-\frac{\displaystyle \int_\Omega F(x,u_j(x))\,
dx}{\|u_j\|_{X_0}^2}\to 0,$$ which, together with (here with $\nu=2$) and , implies that $$\label{F2}
\frac{\displaystyle \int_\Omega F(x,u_j(x))\,
dx}{\|u_j\|_{X_0}^2}\to \frac 1 2$$ as $j\to +\infty$.

Hence, as a consequence of , and , there exists $C>0$ such that $$\label{q2}
\|u_j\|_{L^q(\Omega)}^q\leq C\|u_j\|_{X_0}^2 \mbox{ for every $j\in {\mathbb N}$}.$$

Now, let us show that $$\label{fPnorma2}
\frac{\displaystyle \int_\Omega f(x,u_j(x))Pu_j(x)\,dx}{\|u_j\|_{X_0}^2}\to 0$$ as $j\to +\infty$. Indeed, by and the Hölder inequality, we have $$\begin{aligned}
\frac{\displaystyle \int_\Omega |f(x,u_j(x))Pu_j(x)|\,
dx}{\|u_j\|_{X_0}^2} & \leq
\frac{\|Pu_j\|_\infty}{\|u_j\|_{X_0}^2}\left(a_1|\Omega|+a_2\int_\Omega |u_j(x)|^{q-1}\,
dx\right)\\
& \leq \|Pu_j\|_\infty
\left[\frac{a_1|\Omega|}{\|u_j\|_{X_0}^2}+\frac{a_2'}{\|u_j\|_{X_0}^{2/q}}
\left(\frac{\displaystyle\int_\Omega |u_j(x)|^q
dx}{\|u_j\|_{X_0}^2}\right)^{1-1/q}\right]\\
& \leq \|Pu_j\|_\infty
\left[\frac{a_1|\Omega|}{\|u_j\|_{X_0}^2}+\frac{a_2'C^{1-1/q}}{\|u_j\|_{X_0}^{2/q}}\right]\,,
\end{aligned}$$ thanks to . Thus, follows from this, and .

Finally, dividing both sides of by $\|u_j\|_{X_0}^2$, using – and we get $$\frac{\displaystyle{2\int_\Omega F(x,u_j(x))\, dx -\int_\Omega f(x,u_j(x))u_j(x)\,dx}}{\|u_j\|_{X_0}^2} \to 0\,,$$ which, arguing as above, yields $$\frac{\displaystyle{2\int_\Omega F(x,u_j(x))\, dx}}{\|u_j\|_{X_0}^2} \to 0\,,$$ as $j\to +\infty$. Of course, this is in contradiction with . The proof of Lemma \[lemmanabla2\] is complete.

As a consequence of Lemma \[lemmanabla1\] and Lemma \[lemmanabla2\], we get the following result on the validity of the $\nabla$-condition for $\mathcal J_\lambda$.

\[propnabla\] Let $k$ and $m$ in ${\mathbb N}$ be such that $\lambda_{k-1}<\lambda_k=\dots = \lambda_{k+m-1}<\lambda_{k+m}$ and let $f$ be a function satisfying conditions –.

Then, for any $\sigma>0$ there exists $\varepsilon_\sigma>0$ such that for any $\lambda\in [\lambda_{k-1}+\sigma, \lambda_{k+m}-\sigma]$ and for any $\varepsilon', \varepsilon''\in (0, \varepsilon_\sigma)$, with $\varepsilon'<\varepsilon''$, functional $\mathcal J_\lambda$ satisfies the $(\nabla)(\mathcal J_\lambda, \mathbb H_{k-1}\oplus \mathbb P_{k+m-1}, \varepsilon', \varepsilon'')$ condition.

Assume by contradiction that there exists $\sigma>0$ such that for every ${\varepsilon}_0>0$ there exist $\bar \lambda\in [\lambda_{k-1}+\sigma,\lambda_{k+m}-\sigma]$ and ${\varepsilon}'<{\varepsilon}''$ in $(0,{\varepsilon}_0)$ such that $$\label{notnabla}
(\nabla)(\mathcal J_{\bar\lambda}, \mathbb H_{k-1}\oplus \mathbb P_{k+m-1}, \varepsilon', \varepsilon'')\,\,\, \mbox{does not hold}.$$ Associated to such a $\sigma$, take ${\varepsilon}_0>0$ as provided by Lemma \[lemmanabla1\].

By we can find a sequence $\{u_j\}_{j\in {\mathbb N}}$ in $X_0$ such that $$\label{ujadd}
\begin{aligned}
& \mathcal J_{\bar\lambda}(u_j)\in [{\varepsilon}',{\varepsilon}'']\,\,\,  \mbox{for all}\,\, j\in {\mathbb N}\,,\\
& dist(u_j,\mathbb H_{k-1}\oplus \mathbb P_{k+m-1})\to 0\\
& P_{\mathbb H_{k-1}\oplus \mathbb P_{k+m-1}}\nabla \mathcal J_{\bar\lambda}(u_j)\to 0\,\,\, \mbox{in}\,\, X_0
\end{aligned}$$ as $j\to +\infty$.

By Lemma \[lemmanabla2\] we know that $\{u_j\}_{j\in {\mathbb N}}$ is bounded in $X_0$, and so we can assume that for some $u_\infty\in X_0$ $$\label{convujadd}
\begin{aligned}
& u_j\rightharpoonup u_\infty\,\,\, \mbox{in}\,\, X_0\\
& u_j\to u_\infty\,\,\, \mbox{in}\,\, L^\nu(\Omega)\,\,\, \mbox{for any}\,\, \nu\in[1, 2^*)\\
& u_j\to u_\infty\,\,\, \mbox{a.e. in}\,\, \Omega
\end{aligned}$$ as $j\to +\infty$ and for any $\nu\in [1, 2^*)$ there exists $\ell_\nu\in L^\nu({\mathbb R}^n)$ such that $$\label{dominata20addbis}
|u_j(x)|\leq \ell_\nu(x) \quad \mbox{a.e. in}\,\, {\mathbb R}^n\,\quad \mbox{for any}\,\,j\in {\mathbb N}\,.$$

Now, note that by we can write $$\label{add55}
\begin{aligned}
P_{\mathbb H_{k-1}\oplus \mathbb P_{k+m-1}} \nabla \mathcal J_{\bar\lambda}(u_j) & =u_j
-P_{\footnotesize{\mbox{span}}\left\{e_k,\ldots,\,e_{k+m-1}\right\}}u_j\\
& \qquad \qquad \qquad -P_{\mathbb H_{k-1}\oplus \mathbb P_{k+m-1}}\mathcal L_K^{-1}(\bar\lambda u_j+f(x,u_j)).
\end{aligned}$$ Hence, recalling that $\mathcal L_K^{-1}:L^{q'}(\Omega)\to X_0$ is a compact operator (see Section \[secgrad\]), and that $f(x,u_j)\to f(x,u_\infty)$ in $L^{q'}(\Omega)$ by - and , we get that $$P_{\mathbb H_{k-1}\oplus \mathbb P_{k+m-1}}\mathcal L_K^{-1}(\bar \lambda
u_j+f(x,u_j))\to P_{\mathbb H_{k-1}\oplus \mathbb P_{k+m-1}}\mathcal L_K^{-1}(\bar \lambda
u_\infty+f(x,u_\infty))$$ as $j\to +\infty$ and so, taking into account , and , we deduce that $$\label{ujforte}
u_j\to P_{\mathbb H_{k-1}\oplus \mathbb P_{k+m-1}}\mathcal L_K^{-1}(\bar \lambda
u_\infty+f(x,u_\infty))=:u_\infty\,\,\, \mbox{in}\,\, X_0$$ as $j\to +\infty$.

Furthermore, again by we have that $$\langle \nabla \mathcal J_{\bar \lambda}(u_j), \varphi \rangle_{X_0} \to 0\,\,\, \mbox{for any}\,\, \varphi \in \mathbb H_{k-1}\oplus \mathbb P_{k+m-1}\,,$$ that is, taking into account , $$\label{ujcritico}
\langle \mathcal J'_{\bar\lambda}(u_j), \varphi\rangle= \langle u_j, \varphi\rangle_{X_0}
-\lambda \int_\Omega u_j(x)\varphi(x)\,dx-\int_\Omega f(x, u_j(x))\varphi(x)\,dx \to 0$$ as $j\to +\infty$. Thus, by , , and , we obtain that $$\langle \mathcal J'_{\bar\lambda}(u_\infty), \varphi\rangle= \langle u_\infty, \varphi\rangle_{X_0}
-\lambda \int_\Omega u_\infty(x)\varphi(x)\,dx-\int_\Omega f(x, u_\infty(x))\varphi(x)\,dx$$ for any $\varphi \in \mathbb H_{k-1}\oplus \mathbb P_{k+m-1}$, i.e. $u_\infty$ is a critical point of $\mathcal J_{\bar\lambda}$ constrained on $\mathbb H_{k-1}\oplus \mathbb P_{k+m-1}$.

Hence, Lemma \[lemmanabla1\] yields that $u_\infty\equiv 0$. However, $0<{\varepsilon}'\leq \mathcal J_{\bar\lambda}(u_j)$ for every $j\in {\mathbb N}$, so that, by continuity of $\mathcal J_{\bar\lambda}$, we find $\mathcal J_{\bar\lambda}(u_\infty)>0$, which is absurd. This completes the proof of Proposition \[propnabla\].

Proof of main theorem {#sec:proofthmain}
=====================

This section is devoted to the proof of main result of the paper, concerning the existence of multiple solutions for problem . In order to get this result we apply the following abstract critical point theorem ([@marsac:sns Theorem 2.10]):

\[thmarinosaccon\] Let $H$ be a Hilbert space and $X_1, X_2, X_3$ be three subspaces of $H$ such that $H=X_1\oplus X_2 \oplus X_3$ with $0<\mbox{dim}\,X_i<\infty$ for $i=1,2$. Let $\mathcal I:H\to {\mathbb R}$ be a $C^{1,1}$ functional. Let $\rho, \rho', \rho'', \rho_1$ be such that $0<\rho_1$, $0\leq \rho'<\rho<\rho''$ and $$\Delta=\{u\in X_1\oplus X_2 : \rho'\leq \|P_2u\|\leq \rho'', \|P_1u\|\leq \rho_1\}\,\,\, \mbox{and}\,\,\,T=\partial_{X_1\oplus X_2}\Delta\,,$$ where $P_i:H\to X_i$ is the orthogonal projection of $H$ onto $X_i$, $i=1,2$, and $$S_{23}(\rho)=\{u\in X_2\oplus X_3 : \|u\|=\rho\}\,\,\, \mbox{and}\,\,\,B_{23}(\rho)=\{u\in X_2\oplus X_3 : \|u\|<\rho\}\,.$$ Assume that $$a'=\sup \mathcal I(T)<\inf \mathcal I(S_{23}(\rho))=a''\,.$$ Let $a,b$ be such that $a'<a<a''$, $b>\sup \mathcal I(\Delta)$ and $$\mbox{the assumption}\,\,\, (\nabla)(\mathcal I, X_1\oplus X_3, a, b)\,\,\, \mbox{holds};$$ $$\mbox{the Palais--Smale condition holds at any level}\,\,\, c\in [a,b]\,.$$ Then, $\mathcal I$ has at least two critical points in $\mathcal I^{-1}([a,b])$.

If, furthermore, $$a_1<\inf \mathcal I(B_{23}(\rho))>-\infty$$ and the Palais–Smale condition holds at every $c\in [a_1, b]$, then $\mathcal I$ has another critical level between $a_1$ and $a'$.

First of all, we need the following result:

\[limite\] Let $k$ and $m$ in ${\mathbb N}$ be such that $\lambda_{k-1}<\lambda<\lambda_k=\dots = \lambda_{k+m-1}<\lambda_{k+m}$ and let $f$ satisfy –.

Then, the following relation is verified: $$\lim_{\lambda \to \lambda_k}\sup_{u\in \mathbb H_{k+m-1}} \mathcal J_\lambda(u)=0.$$

First of all, note that $\mathcal J_\lambda$ attains a maximum in $\mathbb H_{k+m-1}$ by .

Now, assume by contradiction that there exist $\{\mu_j\}_{j\in {\mathbb N}}$, such that $$\label{muj}
\mu_j\to \lambda_k$$ as $j\to +\infty$, $\{u_j\}_{j\in {\mathbb N}}$ in $\mathbb H_{k+m-1}$ and ${\varepsilon}>0$ such that for any $j\in {\mathbb N}$ $$\label{sup}
\mathcal J_{\mu_j}(u_j)=\sup_{u\in \mathbb H_{k+m-1}} \mathcal J_{\mu_j} (u)\geq {\varepsilon}.$$

If $\{u_j\}_{j\in {\mathbb N}}$ is bounded in $X_0$, we can assume that $u_j\to u_\infty$ in $X_0$ as $j\to +\infty$ for some $u_\infty \in \mathbb H_{k+m-1}$. Then, by , and the fact that $\mathbb H_{k+m-1}$ is finite-dimensional, we have that $$\mathcal J_{\mu_j}(u_j)\to \mathcal J_{\lambda_k}(u_\infty)$$ as $j\to +\infty$, and so by , and , we immediately get $$\begin{aligned}
{\varepsilon}\leq \mathcal J_{\lambda_k}(u_\infty) & =\frac{1}{2}\|u_\infty\|_{X_0}^2-\frac{\lambda_k}{2}\int_\Omega
|u_\infty(x)|^2\, dx -\int_\Omega F(x,u_\infty(x))\, dx\\
& \leq \frac 1 2 (\lambda_{k+m-1}-\lambda_k)\int_\Omega
|u_\infty(x)|^2\, dx -\int_\Omega F(x,u_\infty(x))\, dx\leq 0,
\end{aligned}$$ and a contradiction arises.

Otherwise, if $\{u_j\}_{j\in{\mathbb N}}$ is unbounded in $X_0$, we can suppose that $$\label{ujadd66}
\|u_j\|_{X_0}\to +\infty$$ as $j\to +\infty$. Therefore, and imply $$0<{\varepsilon}\leq \mathcal J_{\mu_j}(u_j)\leq \frac{1}{2}\|u_j\|_{X_0}^2-
\frac{\mu_j}{2}\int_\Omega |u_j(x)|^2\, dx-a_3\int_\Omega |u_j(x)|^q dx+a_4|\Omega|.$$ But all norms are equivalent in $\mathbb H_{k+m-1}$, so that the right hand side of the previous inequality tends to $-\infty$ as $j\to +\infty$, since $q>2$ by assumption, and holds true. Hence, a contradiction arises as well.

Now, we can prove our multiplicity result for problem . The idea consists in applying Theorem \[thmarinosaccon\] to $\mathcal J_\lambda$, in connection with a classical Linking Theorem (see [@rabinowitz Theorem 5.3]). First we prove:

\[2soluzioni\] Let $k$ and $m$ in ${\mathbb N}$ be such that $\lambda_{k-1}<\lambda<\lambda_k=\dots = \lambda_{k+m-1}<\lambda_{k+m}$ and let $f$ satisfy –.

Then, there exists a left neighborhood $\mathcal O_k$ of $\lambda_k$ such that for all $\lambda \in \mathcal O_k$, problem  has two nontrivial solutions $u_i$ such that $$0<\mathcal J_\lambda(u_i)\leq \sup_{u\in \mathbb H_{k+m-1}} \mathcal J_\lambda(u)$$ for $i=1,2$.

The strategy consists in applying Theorem \[thmarinosaccon\] to the functional $\mathcal J_\lambda$. For this purpose, fix $\sigma>0$ and find ${\varepsilon}_\sigma$ as in Proposition \[propnabla\]. Then, for all $\lambda\in [\lambda_{k-1}+\sigma, \lambda_{k+m}-\sigma]$ and for every ${\varepsilon}',{\varepsilon}''\in(0,{\varepsilon}_\sigma)$, functional $\mathcal J_\lambda$ satisfies the $(\nabla)(\mathcal J_\lambda, \mathbb H_{k-1}\oplus \mathbb P_{k+m-1}, \varepsilon', \varepsilon'')$ condition.

By Lemma \[limite\] there exists $\sigma_1\leq
\sigma$ such that, if $\lambda \in (\lambda_k-\sigma_1,\lambda_k)$, then $$\label{epsilonsup}
\sup_{u\in \mathbb H_{k+m-1}} \mathcal J_\lambda(u)<{\varepsilon}''\,.$$ Moreover, since $\lambda<\lambda_k$, Proposition \[propgeometria\] holds true. Also, $\mathcal J_\lambda$ satisfies the Palais–Smale condition at any level, by Proposition \[lemmaPS\].

Then, by Theorem \[thmarinosaccon\], there exist two critical points $u_1,u_2$ of $\mathcal J_\lambda$ with $$\label{epsilon}
\mathcal J_\lambda(u_i)\in [{\varepsilon}',{\varepsilon}'']\,,$$ $i=1,2$. In particular $u_1$ and $u_2$ are non-trivial solutions of problem  such that $$0<\mathcal J_\lambda(u_i)\leq \sup_{u\in \mathbb H_{k+m-1}} \mathcal J_\lambda(u)\,\,\,\,\, i=1,2\,,$$ since $\varepsilon''$ is arbitrary in and , and this ends the proof of Proposition \[2soluzioni\].

We are now ready to conclude with the

By the classical Linking Theorem (see [@rabinowitz Theorem 5.3]), for any $\lambda\in (\lambda_{k-1}, \lambda_k)$ one can prove the existence of a solution $u_3$ of problem  with $$\label{stimasup}
\mathcal J_\lambda(u_3)\geq \inf_{u\in \mathbb P_{k-1},\, \|u\|=\varrho} \mathcal J_\lambda(u)\geq \beta,$$ for suitable $\varrho>0$ and $\beta>0$, see [@svlinking].

By Lemma \[limite\], we can choose $\lambda$ so close to $\lambda_k$ that $$\label{confr}
\sup_{u\in \mathbb H_{k+m-1}} \mathcal J_\lambda (u)< \inf_{u\in \mathbb P_{k-1},\, \|u\|=\varrho} \mathcal J_\lambda(u)\,.$$

Hence, inequalities , and Proposition \[2soluzioni\] immediately imply that $$\mathcal J_\lambda(u_i)\leq \sup_{u\in \mathbb H_{k+m-1}} \mathcal J_\lambda(u)<\mathcal J_\lambda(u_3)$$ and so $u_3\neq u_i,$ $i=1,2$. The proof of Theorem \[thmain\] is complete.

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[^1]: The first and the third author were supported by the INdAM-GNAMPA Project 2015 [*Modelli ed equazioni non-local di tipo frazionario*]{}. The second author is a member of GNAMPA and is supported by the MIUR National Research Project [*Variational and perturbative aspects of nonlinear differential problems*]{}. The third author was supported by the MIUR National Research Project [*Variational and Topological Methods in the Study of Nonlinear Phenomena*]{} and by the ERC grant $\epsilon$ ([*Elliptic Pde’s and Symmetry of Interfaces and Layers for Odd Nonlinearities*]{}).
