Locating the peaks of least-energy solutions to a quasilinear elliptic
  Neumann problem
LOCATING THE PEAKS OF LEAST-ENERGY SOLUTIONS TO
A QUASILINEAR ELLIPTIC NEUMANN PROBLEM
YI LI AND CHUNSHAN ZHAO
Abstract. In this paper we study the shape of least-energy solutions to the
quasilinear problem εm∆mu−u
m−1 + f (u) = 0 with homogeneous Neumann
boundary condition. We use an intrinsic variation method to show that as
ε → 0+, the global maximum point Pε of least-energy solutions goes to a
point on the boundary ∂Ω at the rate of o(ε) and this point on the boundary
approaches to a point where the mean curvature of ∂Ω achieves its maximum.
We also give a complete proof of exponential decay of least-energy solutions.
1. Introduction and statement of results
In this paper we study the shape of certain solutions to the following quasilinear
elliptic Neumann problem:
(1.1)
εm∆mu− um−1 + f (u) = 0, u > 0 in Ω,
= 0 on ∂Ω,
where m (2 ≤ m < N) and 0 < ε ≤ 1 are constants and Ω ⊆ RN (N ≥ 3)
is a smooth bounded domain. The operator ∆mu = div(|∇u|m−2 ∇u) is the m-
Laplacian operator, and ν is the unit outer normal to ∂Ω.
Problem (1.1) appears in the study of non-Newtonian fluids, chemotaxis and
biological pattern formation. For example, in the study of non-Newtonian fluids,
the quantity m is a characteristic of the medium: media with m > 2 are called
dilatant fluids, and those with m < 2 are called pseudo-plastics. If m = 2, they
are Newtonian fluids (see [3] and its bibliography for more backgrounds). For
the case m = 2, (1.1) is also known as the stationary equation of the Keller–
Segal system in chemotaxis [14] or the limiting stationary equation of the so-called
Gierer–Meinhardt system in biological pattern formation (see [23]).
First let us recollect some results related to our problem. In a series of remarkable
papers, C.-S. Lin, W.-M. Ni and I. Takagi [14], Ni and Takagi [17], [18] studied the
Neumann problem for certain elliptic equations, including
(1.2)
d∆u− u+ up = 0, u > 0 in Ω,
= 0 on ∂Ω,
where d > 0, p > 1 are constants, and p is subcritical, i.e., p < N+2
. First, Lin,
Ni and Takagi [14] applied the mountain-pass lemma [1] to show the existence of
Key words and phrases. Quasilinear Neumann problem, m-Laplacian operator, least-energy
solution, exponential decay, mean curvature.
http://arxiv.org/abs/0704.0402v1
2 YI LI AND CHUNSHAN ZHAO
a least-energy solution ud to (1.2), by which is meant that ud has the least energy
among all solutions to (1.2) with the energy functional
Id (u) =
|∇u|2 + 1
u2 − 1
defined on W 1,2 (Ω). Hereinafter u+ = max {u, 0} and u− = min {u, 0}. Then in
[17], [18], Ni and Takagi investigated the shape of the least-energy solution ud as
d becomes sufficiently small, and showed that ud has exactly one peak (i.e., local
maximum of ud) at Pd ∈ ∂Ω. Moreover, as d tends to zero, Pd approaches a point
where the mean curvature of ∂Ω achieves its maximum. See [15] for a review in
this field. Also see [16] for the critical case p = N+2
, and [5], [6], [7], [8], [9] for
existence and properties of multiple-peaks solutions to (1.2).
From now on we make some hypotheses on f : R → R, as follows.
(H2) f (t) ≡ 0 for t ≤ 0 and f ∈ C1 (R).
(H3) f(t) = O (t
p) as t→ ∞ with m− 1 < p <
N (m− 1) +m
N −m .
(H4) Let F (t) =
f (s) ds. Then there exists a constant θ ∈
that F (t) ≤ θtf (t) for t > 0.
f (t)
is strictly increasing for t > 0 and f (t) = O
tm−1+δ
as t→ 0+ with
a constant δ > 0.
(H6) Let g (u) =
(m− 1)um−1 − uf ′ (u)
um−1 − f (u)
. Then g (u) is non-increasing on (uc,∞),
where uc is the unique positive solution for f (t) = t
Next we present some preliminary knowledge about least energy solutions of the
following problem:
(1.3)
∆mu− um−1 + f(u) = 0 in RN
u > 0 in RN
As before we define an “energy functional” I:W 1,m(RN ) −→ R associated with
(1.3) by
(1.4) I(ṽ) =
(εm |∇ṽ|m + |ṽ|m)− F (ṽ+)
Next let us give a remark on ground states to the problem 1.3. Here by a
ground state we mean a non-negative nontrivial C1 distribution solution which
tends to zero at ∞. For case m = 2, it is well known that the problem 1.3 has
a unique ground state (up to translations) which is radially symmetric [4]. For
case 2 < m < N uniqueness and radial symmetry of ground states are still open.
But the Steiner symmetrization tells us the least-energy solutions must be radially
symmetric (certainly least-energy solutions are ground states). Our assumptions
guarantee that the uniqueness (up to translations) of radial ground states (see
[20]), which implies the uniqueness of least-energy solutions of the problem (1.3).
Exact exponential decay of radial ground states was given in [11], thus we have the
following proposition about the unique radial least-energy solution to problem 1.3:
Proposition 1.1. Under assumptions (H2)–(H6), there is a unique least energy
solution w(x) for (1.3) satisfying:
LOCATING THE PEAKS OF LEAST-ENERGY SOLUTIONS 3
( i ) w is radial, i.e., w(x) = w(|x|) = w(r) and w ∈ C1(RN ) with
w(0) = maxX∈RN w(x), w
′(0) = 0 and w′(r) < 0, ∀r > 0.
(ii) limr−→∞ w(r)r
m(m−1) e(
m r = C0 > 0 for some constant C0 and
limr−→∞
w′(r)
Remark 1.1. A good example for f (t) which satisfies all hypotheses (H2)–(H6) is
f (t) = tp for m− 1 < p < N (m− 1) +m
Next we define an “energy functional” Jε : W
1,m (Ω) → R associated with (1.1)
(1.5) Jε (v) =
(εm |∇v|m + |v|m)− F (v+)
with F (v+) =
f (s) ds. Then the well-known mountain-pass lemma [1] implies
(1.6) cε = inf
t∈[0,1]
Jε (h (t))
is a positive critical value of Jε, where Γ is the set of all continuous paths joining
the origin and a fixed nonzero element e ∈ W 1,m (Ω) such that e ≥ 0 and Jε (e) ≤ 0.
It turns out cε can also be characterized as follows:
cε = inf
Jε (u)
u ∈ W 1,m (Ω) ; u ≥ 0, u 6≡ 0,
(εm |∇u|m + um) dx =
f (u)u dx
(1.7) cε = inf
M [u] | u ∈ W 1,m (Ω) , u 6≡ 0 and u ≥ 0 in Ω
M [u] = sup
Jε (tu) .
Hence cε is the least positive critical value and a critical point uε of Jε with critical
value cε is called a least-energy solution. Notice also that if we let
c∗ = I(w) =
(|∇w|m + wm) dx−
F (w) dx,
where w is the unique least energy solution of (1.3), then c∗ can also be characterized
(1.8) c∗ = inf
M∗ [v] | v ∈ W 1,m
, v 6≡ 0 and v ≥ 0 in RN
M∗ [v] = sup
I (tv) .
We refer to Lemma 2.1 of [13] for the above characterizations.
Next we consider the following problem:
4 YI LI AND CHUNSHAN ZHAO
v ∈W 1,m
with RN+ =
(x1, · · · , xN ) ∈ RN , xN ≥ 0
and satisfies
(1.9)
∆mv − vm−1 + f (v) = 0, v > 0 in RN+ ,
= 0 on xN = 0.
The solutions of (1.9) can be characterized as critical points of the functional defined
over W 1,m
as follows.
(ṽ) =
|∇ṽ|m + ṽm) dx−
F (ṽ+) dx.
Similarly as above the least positive critical value C∗ corresponding to least energy
solutions of (1.9) can be characterized as
(1.10) C∗ = inf
ṽ∈W 1,m(RN+ ),ṽ≥0,ṽ 6≡0
(tṽ)
and moreover
(1.11) C∗ =
due to the boundary condition in (1.9) and the fact that w is radial and hence
= 0. We also refer to Lemma 2.1 of [13] for the above characterization of C∗.
In Theorem 1.3 of [13], we proved the following theorem.
Theorem 1.1. Under hypotheses (H2)–(H6), let uε be a least-energy solution of
(1.1). Then all local maximum points(if more than one) of uε aggregate to a global
maximum point Pε at a rate of o(ε) and dist(Pε, ∂Ω)/ε→ 0 as ε → 0+, where
dist(·, ·) is the general distance function. Moreover, we have the following upper-
bound estimate for cε as ε→ 0+:
(1.12) cε ≤ εN
c∗ − (N − 1) max
H (P ) γε+ o (ε)
where H (P ) denotes the mean curvature of ∂Ω at P , γ > 0 is a positive constant
given by
(1.13) γ =
N + 1
|w′ (|z|)|m zN dz.
Our goal in this paper is to locate the position on ∂Ω where the global maximum
point Pε of uε in Ω approaches, provided ε is sufficiently small. For the case m = 2,
Ni and Takagi [18] located the peak by linearizing the equation d∆u−u+f (u) = 0
around the ground state w. But this method fails for our problem with m 6= 2 due
to the strong nonlinearity of the m-Laplacian operator ∆mu = div(|∇u|m−2 ∇u).
So we have to use the intrinsic variational method created by Del Pino and Felmer
in [2] to attack it. We also give a complete proof of the exponential decay of
the least-energy solution uε. We remark that our proof is complete and does not
require the non-degeneracy of the unique radial least energy solution w as stated
in Proposition 1.1, and hence it is different from Ni’s and Takagi’s work [17]. Now
our results can be stated as follows:
Theorem 1.2. Under hypotheses (H2)–(H6), let uε be a least-energy solution of
(1.1) and P̃ε ∈ ∂Ω with dist(Pε, P̃ε) = dist(Pε, ∂Ω). Then as ε→ 0+, after passing
to a sequence P̃ε approaches P̄ ∈ ∂Ω with
LOCATING THE PEAKS OF LEAST-ENERGY SOLUTIONS 5
(ii) H
= max
H (P ), where H (P ) denotes the mean curvature of ∂Ω at P
as stated before, and moreover
(iii) the associated critical value cε can be estimated as ε→ 0+ as follows :
(1.14) cε = ε
c∗ − (N − 1)H
γε+ o (ε)
where c∗, γ are as stated in Theorem 1.1.
The organization of this paper is as follows: In Section 2, we will prove some
lemmas which will be used in proving Theorem 1.2. The proof of Theorem 1.2 will
be given in Section 3.
2. Some lemmas and exponential decay of uε
First we prove the following lemma related to exponential decay of the least-
energy solution uε.
Lemma 2.1. Let ε be sufficiently small and that the least-energy solution uε
achieves its global maximum at some point Pε. Then there exist two positive con-
stants c3 and c4 independent of uε or ε such that
(2.1)
uε (x) ≤ c3 exp {−c4 |x− Pε| /ε}
|∇uε(x)| ≤ c3ε−1 exp{−c4|x− Pε|/ε}.
Before beginning to prove this lemma, we give a remark on it.
Remark 2.1. For the case m = 2, under the assumption of non-degeneracy of the
linearized operator ∆− 1 + f ′ (w), where w is the unique ground state of (1.3), Ni
and Takagi [18] showed that uε (x) can be written as
(2.2) uε (x) = w (x) + εφ1 (x) + o (ε)
and φ1 (x) enjoys the exponential-decay property ([18]). Clearly we cannot derive
exponential decay of uε (x) as stated in Lemma (2.1) from (2.2) even though both
w (x) and εφ1 (x) have exponential decay property.
Proof of Lemma 2.1. Since ∂Ω is a smooth compact submanifold of RN , it follows
from the tubular neighborhood theorem [10] that there exists a constant ω (Ω) > 0
which depends only on Ω such that ΩI =
x ∈ Ω, d (x, ∂Ω) < ω (Ω)
is diffeomor-
phic to the inner normal bundle
I = {(x, y) : x ∈ ∂Ω, y ∈ (−ω (Ω) , 0] νx} ,
here νx is the unit outer normal of ∂Ω at x, and the diffeomorphism is defined as
follows: ∀x ∈ ΩI , there exists an unique x̂ ∈ ∂Ω such that d (x, x̂) = d (x, ∂Ω) ,
then Φ∗ : x −→ (x̂,−d (x, x̂) νx̂) . Moreover this diffeomorphism satisfies Φ∗|∂Ω =
Identity. Similarly, let ΩO =
x ∈ RN \ Ω, d(x, ∂Ω) < ω (Ω)
. Then ΩO is diffeo-
morphic to the outer normal bundle
O = {(x, y) : x ∈ ∂Ω, y ∈ [0, ω (Ω)) νx} ,
and the diffeomorphism is given as follows. ∀x ∈ ΩO, there exists an unique
x̄ ∈ ∂Ω such that d(x, x̄) = d (x, ∂Ω) , and then Φ# : x −→ (x̄, d (x, x̄) νx̄)
and Φ#|∂Ω = Identity. Note that (∂Ω)NI is clearly diffeomorphic to (∂Ω)
O via
the following reflection Φ∗ : (∂Ω)
I −→ (∂Ω)
O defined by Φ
∗ ((x, y)) = (x,−y) .
6 YI LI AND CHUNSHAN ZHAO
Therefore, Φ = Φ−1∗ ◦ Φ∗−1 ◦ Φ# : ΩO −→ ΩI is the desired diffeomorphism and
Φ|∂Ω = Identity. Moreover, if we let x = Φ(z) = (Φ1(z), · · · ,ΦN(z)) , z ∈ ΩO,
and z = Ψ(x) = Φ−1(x) = (Ψ1(x), · · · ,ΨN(x)) , x ∈ ΩI , gij =
gij =
(Φ (z)) , we have gij |∂Ω = gij |∂Ω = δij with δij being the Kro-
necker symbol. Denote G =
and A = G− I with I being the N ×N identity
matrix, g(x) = det (gij) and ûε(x) = uε (Φ (x)) for x ∈ ΩO. Then ûε(x) satisfies the
following equations:
εmLûε −
gûm−1ε +
gf (ûε) = 0, ûε > 0 in ΩO
= 0, on ∂Ω,
where
Lûε =
s,l=1
∇ûεG (∇ûε)T
g (∇ûε)G
where Tr means taking the trace of a square matrix.
For 0 < γ̃ ≤ ω(Ω), let ΩOγ̃ =
x ∈ ΩO, d(x, ∂Ω) < γ̃
. We know ‖A‖C0 can be
made arbitrarily small by making γ̃ sufficiently small. Next we define
ūε =
uε(x), x ∈ Ω
ûε(x), x ∈ ΩO,
g̃ij =
δij , x ∈ Ω
gij , x ∈ ΩO,
g̃ij =
δij , x ∈ Ω
gij , x ∈ ΩO,
and Ã(x, ξ) =
Ã1(x, ξ), · · · , ÃN (x, ξ)
for ξ = (ξ1, · · · , ξN ) with
Ãi(x, ξ) =
s,l=1
g̃slξsξl
g̃ijξj
and g̃ = det(g̃ij), B(x, u) =
−um−1 + f(u)
. Then ūε(x) satisfies
(2.3) εm div
Ã(x,∇ūε)
+B(x, ūε) = 0 in Ω
in the weak sense.
LOCATING THE PEAKS OF LEAST-ENERGY SOLUTIONS 7
For any ball Br(x0) ⊂ Ω
ΩO with radius r and center x0 ∈ Ω, let ρ = |x− x0|.
Then for any smooth increasing function φ = φ(ρ) we have
(∇φ)G(∇φ)T
g(∇φ)G
∣∇φ(I +A)(∇φ)T
det(I +A)−1∇φ(I +A)
= |∇φ|m−2 ∇φ+
∣∇φ(I + tA)(∇φ)T
det(I + tA)−1∇φ(I + tA)
= |∇φ|m−2 ∇φ
∣∇φ(I + tA)(∇φ)T
(∇φ)A(∇φ)T
det(I + tA)−1∇φ(I + tA)dt
∣∇φ(I + tA)(∇φ)T
det(I + tA)−1
det(I + tA)−1
∇φ(I + tA)dt
∣∇φ(I + tA)(∇φ)T
det(I + tA)−1(∇φ)Adt.
Therefore
(2.4)
(∇φ)G(∇φ)T
g(∇φ)G
3(N − 1)
φ′ +K
by taking γ̃ sufficiently small, here K > 0 is a constant depending only on Ψ, hence
only on Ω and φ′ =
dφ(ρ)
From now on γ̃ = γ̃(Ω) is fixed such that (i) 3
≤ √g ≤ 5
, (ii) (2.4) holds for
any smooth increasing radial function φ(ρ) and (iii) 3
|ξ|m ≤ Ã(x, ξ) · ξ ≤ 5
|ξ|m for
any ξ = (ξ1, · · · , ξN ). Denote Ωγ̃ = Ω ∪ ΩOγ̃ .
Let Ωε =
(Ω− Pε) and uε(x) = uε(Pε + εx) for x ∈ Ωε. Then uε is a solution
to the following problem:
(2.5)
ε − (uε)m−1 + f(uε) = 0, uε > 0 in Ωε
= 0, on ∂Ωε,
where n is the unit outer normal of ∂Ωε. Similarly, let Ω
Ωγ̃ − Pε
ūε(x) = ūε(Pε + εx) for x ∈ Ωγ̃ε . Since ūε converges to the unique radial least-
energy solution w of (1.3) in C1loc(R
N ) ∩W 1,m(RN ) as ε → 0+ (see the proof of
Theorem 1.2 of [13]) and w satisfies:
(i) w is radial, i.e.,w(x) = w(|x|) = w(r) > 0
(ii) lim
w(r)r
m(m−1) e(
m−1 )
m r = C0 > 0
(see Theorem 1 of [11]) which yields w(r) ≤ κe−µr for a constant κ > 0 and
. First we fix a constant η > 0 such that 1
tm−1 > f(t) for t ∈
(0, η]. From hypothesis (H5) it follows that such an η exists. Then there exist
ε0 > 0 sufficiently small and R0 sufficiently large such that 4κ exp{−µR0} < η and
8 YI LI AND CHUNSHAN ZHAO
‖ūε − w‖C0(BR0(0)∩Ωε) ≤ κ exp{−µR0}, which yields
uε|(∂BR0 (0))∩Ωε ≤ 2κ exp{−µR0}.
Note that
ε − 7
(uε)m−1 = 1
(uε)m−1 − f(uε) > 0 in Ωε \BR0(0),
= 0 on ∂Ωε \BR0(0),
uε ≤ 2κ exp{−µR0} on ∂BR0(0) ∩ Ωε.
Then we have
uε(x) ≤ 2κ exp{−µR0}, for x ∈ Ωε \BR0(0)
due to the strong maximum principle ([22]). We get by scaling back that
uε|Ω\BεR0 (0) ≤ 2κ exp{−µR0}
(2.6)
uε(x) ≤ w
+ κ exp{−µR0} ≤ κ exp{−
}+ κ exp{−µR0}
≤ 2κ exp{−µ|x|
for x ∈ Ω ∩BεR0(0).
From definition of ūε we know
ūε(x) ≤ 2κ exp{−
µ (|x| − 2dist (Pε, ∂Ω))
} ≤ 4κ exp{−µ|x|
} for x ∈ Ωγ̃∩BεR0(0)
for ε ∈ (0, ε0] with ε0 sufficiently small due to the fact dist(Pε, ∂Ω) = o(ε) as
ε→ 0+. Note that
Ωγ̃\BεR0 (0)
ūε ≤ 4κ exp{−µR0}.
Choice of R0 and γ̃ tells us for any 0 < t ≤ 4κ exp{−µR0}
B(x, t) =
−tm−1 + f(t)
tm−1.
∀x0 ∈ Ω \BεR0(0) and Br(x0) ⊂ Ωγ̃ \BεR0(0), define
φ(x) = φ(ρ) = φ(|x− x0|)
Ωγ̃\BεR0 (0)
where λ∗ > 0 is a constant to be determined later. Simple calculations show that
(i) φ′(ρ) > 0 and
3(N−1)
φ′ +K
(m− 1) (λ∗)m
))m−2
tanh(λ∗ρε )
( λ∗ρε )
+ε (λ∗)
))m−1
LOCATING THE PEAKS OF LEAST-ENERGY SOLUTIONS 9
for any 0 < λ∗ ≤ λ̂, where λ̂ > 0 is a small constant depending only on m and
Ω through K. We remark that we have used the fact maxr∈[0,∞)
tanhr
< ∞ for
m ≥ 2. From now on we choose λ∗ = λ̂.
Therefore we have
εm div(Ã(x,∇ūε))−
ūm−1ε ≥ 0 in Br(x0),
εm div(Ã(x,∇φ))−
m−1 ≤ 0 in Br(x0).
Clearly
φ|∂Br(x0) ≥ ūε|∂Br(x0).
Then from the Comparison Theorem (Theorem 10.1 of [19]) it follows that
φ(x) ≥ ūε(x) in Br(x0).
In particular, φ(x0) ≥ ūε(x0). Thus we get
uε(x0) ≤
Ωγ̃\BεR0(0)
exp{−λ∗r
Choosing r = d
x0, ∂
Ωγ̃ \BεR0(0)
we get
uε(x0) ≤ 4κ exp{−µR0 −
} ≤ 2κ exp{− λ̃(εR0 + r)
with λ̃ = min{µ, λ∗}. Note that x0 belongs to one of the following two cases:
(i) d
x0, ∂
Ωγ̃ \BεR0(0)
= d (x0, ∂BεR0(0)) ,
(ii) d
x0, ∂
Ωγ̃ \BεR0(0)
x0, ∂Ω
For case (i) we have d(x0, Pε) ≤ εR0 + r and therefore
(2.7) uε(x0) ≤ 4κ exp{−
λ̃d(x0, Pε)
For case (ii) we have r ≥ γ̃ and thus
(2.8)
uε(x0) ≤ 4κ exp{−λ̃
εR0 + r
} ≤ 4κ exp{− λ̃γ̃
≤ 4κ exp{−λ̃ γ̃
diam(Ω)
· d(x0, Pε)
Combining (2.6), (2.7) and (2.8) together and letting c̃3 = 4κ, c̃4 = min{µ, λ̃, λ̃γ̃diam(Ω)}
yields
(2.9) uε(x) ≤ c̃3 exp{−
c̃4|x− Pε|
Next we show the estimate for |∇uε| holds. First from (2.5) it follows that
(2.10) ∆mu
ε = (uε)
m−1 − f(uε), uε > 0 in Ωε
For x ∈ Ωε and dist(x, ∂Ωε) ≥ 1, consider (2.10) in the unit ball centered at x, i.e.,
B1(x). Then by an C
1,α estimate (see [21], for example) there exists two constants
10 YI LI AND CHUNSHAN ZHAO
C > 0 and α∗ ∈ (0, 1) which are independent of ε such that
(2.11)
‖uε‖C1,α∗ (B 1
(x)) ≤ C
‖uε‖L∞(B1(x)) + ‖ (u
m−1 − f(uε)‖
L∞(B1(x))
≤ c∗3 exp{−c∗4|x− Pε|},
where we have used (2.9) and the fact that uε(x) = uε(Pε + εx) for x ∈ Ωε.
Especially we have
(2.12) |∇uε(x)| ≤ c∗3 exp{−c∗4|x− Pε|},
for x ∈ Ωε and dist(x, ∂Ωε) ≥ 1. For x ∈ Ωε with dist(x, ∂Ωε) < 1. Let x0 ∈ ∂Ωε
be a point such that dist(x, x0) = dist(x, ∂Ωε) and consider ū
ε(x) = ūε(Pε + εx)
in B2(x0), the ball of radius 2 centered at x0, then from (2.3) it follows that ū
satisfies
(2.13) div
Ã(Pε + εx,∇ūε)
+B(Pε + εx, ū
ε) = 0 in B2(x0)
in the weak sense. Then applying an C1,α estimate (see [21], for example) again
yields as above that there exists two constants C > 0 and α∗ ∈ (0, 1) which are
independent of ε such that
‖ûε‖C1,α∗(B1(x0)) ≤ C
‖ûε‖L∞(B2(x0)) + ‖B(Pε + εx, û
L∞(B2(x0))
≤ c∗3 exp{−c∗4|x− Pε|}
by adjusting c∗3 and c
4 if it is necessary. Especially we have
(2.14) |∇uε(x)| ≤ c∗3 exp{−c∗4|x− Pε|},
Thus combining (2.11) and (2.14) together and scaling back we have for x ∈ Ω
|∇uε(x)| ≤ c∗3ε−1 exp{−c∗4
|x− Pε|
Proof of Lemma 2.1 is completed by letting c3 = max{c̃3, c∗3} and c4 = min{c̃4, c∗4}.
Remark 2.2. Our proof of the Lemma 2.1 with necessary minor modifications also
works well for elliptic systems.
Next we present a lemma related to extensions of uε.
Lemma 2.2. There exists a C1-extension ũε of uε which has compact support in
N and satisfies
(ii) ‖ũε‖W 1,m(RN ) ≤ c5 ‖uε‖W 1,m(Ω) and ‖ũε‖C1(RN ) ≤ c5 ‖uε‖C1(Ω̄),
(iii) ũε also has the exponential-decay property as stated in Lemma 2.1, i.e.,
there exists an absolute constant λ ≥ 1 such that
(2.15)
0 ≤ ũε ≤ c3λ exp
|x− Pε|
|∇ũε(x)| ≤ c3λε−1 exp{−
|x− Pε|
LOCATING THE PEAKS OF LEAST-ENERGY SOLUTIONS 11
(iv) there exists a positive constant δ̃ = δ̃ (Ω) such that for any P ∈ ∂Ω,
ũε|B
(P )\Ω is the reflection of uε through ∂Ω.
Proof. Let d̃ = d
∂Ω, ∂Ωγ̃
and 0 ≤ ̺(x) ≤ 1 be a smooth cut-off function such
that ̺(x) ≡ 1 for x ∈ {x ∈ RN , d(x,Ω) ≤ d̃
} and ̺(x) ≡ 0 for x ∈ RN \
Then ũε = ̺ūε satisfies (ii), (iii) and (iv) automatically. The proof of this lemma
is completed. �
Similar to energy density introduced in [2], we define the energy density associ-
ated with (1.1) as follows:
E (w, y′) =
(|∇w|m + wm)− F (w)
(y′, 0) for y′ ∈ RN−1.
Then we have the following lemma.
Lemma 2.3. Let G be a C2 function in a neighborhood of the origin of RN−1.
i,j=1
Gij (0) yiyjE (w, y
′) dy′ = 2∆G (0) γ,
where γ is the constant defined in (1.13), and y′ = (y1, . . . , yN−1), and
Gij (0) =
∂yi∂yj
(0) .
Proof. In Lemma 2.4 of [13], we showed that
(2.16) γ =
(|∇w|m) + wm − F (w)
zN dz.
Next we introduce the polar coordinates
z1 = r sin θN−1 sin θN−2 · · · sin θ2 sin θ1,
z2 = r sin θN−1 sin θN−2 · · · sin θ2 cos θ1,
z3 = r sin θN−1 sin θN−2 · · · cos θ2,
... ,
zN = r cos θN−1,
and notice that
(r, θ1, . . . , θN−1) | r > 0, 0 ≤ θ1 < 2π,
0 ≤ θj < π for j = 2, . . . , N − 2, and 0 ≤ θN−1 <
and that
dz = rN−1 sin θ2 sin
2 θ3 · · · sinN−2 θN−1 dr dθ1 · · · dθN−1.
After elementary computations one obtains
(2.17) γ =
|w′ (r)|m + wm (r)
− F (w (r))
rN dr · ωN−2,
where ωN−2 is the volume of the unit ball in R
N−2. Here we used the fact that w
is radially symmetric.
12 YI LI AND CHUNSHAN ZHAO
Using the radial symmetry of w again, we obtain
i,j=1
Gij (0) yiyjE (w, y
′) dy′(2.18)
Gii (0) y
iE (w, y
′) dy′
Gii (0) ·
N − 1
|y′|2E (w, y′) dy′
= ∆G (0) ·
E (w, r) rN dr · ωN−2,
where E (w, r) = (1/m)
|w′ (r)|m + wm (r)
− F (w (r)) . Comparing (2.17) and
(2.18) yields
i,j=1
Gij (0) yiyjE (w, y
′) dy′ = 2∆G (0) γ.
The proof of Lemma 2.3 is completed. �
3. Proof of Theorem 1.2
With the help of the lemmas in Section 2, now we can give the proof of Theorem
Proof of Theorem 1.2. Since as ε→ 0+, Pε → ∂Ω at the rate of o(ε), it follows that
d(Pε, P̃ε)/ε → 0, where P̃ε ∈ ∂Ω is the closest point on ∂Ω to Pε. then by passing
to a sequence, P̃ε → P̄ ∈ ∂Ω. After an ε-dependent rotation and translation,
we may assume that P̃ε is at the origin and Ω can be described in a fixed cubic
neighborhood V of P̄ as the set
{ (x′, xN ) | xN > ψε (x′) } with x′ = (x1, . . . , xN−1) ,
where ψε is smooth, ψε (0) = 0, ∇ψε (0) = 0. Furthermore, we may assume that
ψε converges locally in the C
2 sense to ψ, a corresponding parametrization at P̄ .
Note that since P̃ε is the origin, so we have Pε/ε → 0 as ε → 0+. Thus we have
ũε(x) = ũε(εx) = ũε
x− Pε
→ w(x) in C1loc
as ε → 0+. From the
characterization of cε = Jε (uε) in Section 1, we have
ε−NJε (uε) ≥ ε−NJε (tuε) = IΩε (tuε)
for all t > 0. Hereinafter
IΩ∗ (v) =
(|∇v|m + |v|m) dx−
F (v) dx.
IΩε (tu
ε) = IΩε (tũ
ε) ≥ I
(tũε) + I(Ωε∩Vε)\RN+
(tũε)− I(RN+∩Vε)\Ωε (tũ
ε)(3.1)
= I + II− III,
LOCATING THE PEAKS OF LEAST-ENERGY SOLUTIONS 13
with Vε =
V. Let us choose t = tε so that IRN
(tũε) maximizes in t. Then from
the definition of C∗ in (1.10), equality (1.11) and Lemma 2.2 it follows that
I = I
(tεũ
ε) ≥ c∗
e−c6/ε
for some constant c6 > 0 independent of ε. Next we give an estimate of tε.
Lemma 3.1. There is a unique tε ∈ (0,∞) such that
tmε (|∇ũε|
+ (ũε)m) dx −
F (tεũ
ε) dx
= sup
tm (|∇ũε|m + (ũε)m) dx−
F (tũε) dx
and moreover
(3.2) tε = 1 + o (1) as ε→ 0+.
Proof. Under assumption (H5), the existence and uniqueness of tε can be proved
similarly to the proof of Lemma 2.1 of [13]. Here we only need show (3.2). Let
(3.3) hε (t) =
(|∇ũε|m + (ũε)m) dx−
F (tũε) dx.
(3.4)
h′ε (t) = t
(|∇ũε|m + (ũε)m) dx−
ũεf (tũε) dx
= tm−1
(|∇w|m + wm) dx−
wf (tw) dx+ o(1),
here we have used the exponential decay of ũε in Lemma 2.2, exponential decay of
w and ũε → w in C1loc
as ε → 0+. Moreover the term o (1) → 0 uniformly
in t on each compact interval as ε → 0+. (3.3) tells us hε(1) = 12c∗ + o(1), which
yields that tε is bounded and away from 0. Also from (3.4) it follows that
(3.5)
h′ε (t) = t
wf (w) dx−
w f (tw) dx+ o (1)
= tm−1
f (w)
− f (tw)
dx+ o (1) .
Therefore at t = tε we have
(3.6)
f (w)
− f (tεw)
(tεw)
dx = o (1) .
Since f(t)/tm−1 is strictly increasing (see (H5)) it follows from (3.6) that tε =
1 + o (1) . The proof of Lemma 3.1 is completed. �
14 YI LI AND CHUNSHAN ZHAO
Proof of Theorem 1.2 continued. Using again the exponential decay of uε in
Lemma 2.1 and the expansion of tε in Lemma 3.1, we obtain
−II = −
(RN−1×{0})∩Vε
dy′(3.7)
(ψε(εy′))
tmε (|∇ũε|
+ (ũε)
)− F (tεũε)
(y′, yN ) dyN
= − (1 + o (1))
(RN−1×{0})∩(Ωε∩Vε)
(ψε(εy′))
(|∇uε|m + (uε)m)− F (uε)
(y′, yN) dyN .
Similarly,
(3.8) III = (1 + o (1))
Vε∩(RN−1×{0})
(ψε(εy
(|∇ũε|m + (ũε)m)− F (ũε)
(y′, yN) dyN .
In above a+ = max{a, 0}, a− = min{a, 0}. Since ψε (0) = 0, ∇ψε (0) = 0 and
ψε converges in the C
2 local sense to ψ, and ũε → w in the C1 local sense in RN
with uniform exponential decay with respect to ε, it follows from the dominated
convergence theorem that
(−II + III)
i,j=1
ψij (0) yiyj
(|∇w|m + wm)− F (w)
(y′, 0) dy′
= ∆ψ (0) γ = (N − 1)H
γ (by Lemma 2.3).
Thus we have
cε ≥ εN
c∗ − (N − 1)H
γε+ o (ε)
But (1.12) in Theorem 1.1 tells us
cε ≤ εN
c∗ − (N − 1) max
H (P ) γε+ o (ε)
Therefore we get
(ii) H
= max
H (P ), which is (ii) of Theorem 1.2,
(iii) cε = ε
c∗ − (N − 1)H
γε+ o (ε)
as ε→ 0+,
which is part (iii) of Theorem 1.2. The proof of Theorem 1.2 is completed. �
Acknowledgement. The authors want to give their thanks to anonymous
referee for some helpful comments.
LOCATING THE PEAKS OF LEAST-ENERGY SOLUTIONS 15
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Department of Mathematics, The University of Iowa, Iowa City, IA 52242
16 YI LI AND CHUNSHAN ZHAO
Department of Mathematics, Hunan Normal University, Changsha, Hunan
E-mail address: yi-li@uiowa.edu
Department of Mathematical Sciences, Georgia Southern University, Statesboro,
GA 30460
E-mail address: czhao@GeorgiaSouthern.edu
	1. Introduction and statement of results
	2. Some lemmas and exponential decay of u0=x"0122
	3. Proof of Theorem ??
	References
