1 Introduction and Main Result
We study the following fractional Schrödinger equation:
(1.1)
{
ε
2
s
(
-
Δ
)
s
u
+
V
(
x
)
u
=
f
(
u
)
in
ℝ
N
,
u
∈
H
s
(
ℝ
N
)
,
u
>
0
on
ℝ
N
,
where ε is a small positive parameter, N>2s(-Δ)s with s∈(0,1) is the fractional Laplacian and the potential V:ℝN→ℝ is a continuous function satisfying the following conditions:
inf
x
∈
ℝ
N
V
(
x
)
=
α
>
0
.
There is a bounded domain Λ such that
V
0
:=
inf
Λ
V
<
min
∂
Λ
V
,
and we set ℳ:={x∈Λ:V(x)=V0}.
This hypothesis was first introduced by del Pino and Felmer [14]. The nonlinearity f:ℝ→ℝ is a continuous function. Since we are looking for positive solutions, we assume that f(t)=0 for t<0. Furthermore, we need the following conditions:
lim
t
→
0
+
f
(
t
)
/
t
=
0
.
lim
t
→
+
∞
f
(
t
)
/
t
2
s
*
-
1
=
1
, where 2s*=2N/(N-2s).
There exist λ>0 and 2<q<2s* such that f(t)≥λtq-1+t2s*-1 for t≥0.
Note that, for the case s=1, conditions (f1)–(f3) were first introduced by Zhang and Zou [34]. These hypotheses can be regarded as an extension of the celebrated Berestycki–Lions-type nonlinearity (see [5, 6]) to the fractional Schrödinger equations with critical growth.
Remark 1.1.
Without loss of generality, in the present paper we assume that 0∈ℳ.
A basic motivation for the study of (1.1) mainly comes from looking for semiclassical standing wave solutions which are of the form ψ(x,t)=exp(-iEt/ε)u(x) for the following fractional nonlinear Schrödinger equations
(1.2)
i
ε
∂
ψ
∂
t
=
ε
2
s
(
-
Δ
)
s
ψ
+
(
V
(
x
)
+
E
)
ψ
-
f
(
ψ
)
,
(
t
,
x
)
∈
ℝ
×
ℝ
N
,
where ε is a small positive constant which corresponds to the Planck constant, i is the imaginary unit, V(x) is a potential function and f(exp(iθ)ξ)=exp(iθ)f(ξ) for θ,ξ∈ℝ.
Equations of the form (1.2) have been derived as models of many physical phenomena such as phase transition, conservation laws, especially in fractional quantum mechanics, etc.; see [18]. Formula (1.2) was introduced by Laskin [25, 24] as an extension of the classical nonlinear Schrödinger equations s=1 in which the Brownian motion of the quantum paths is replaced by a Lévy flight; we refer to [15] for more physical backgrounds.
Formula (1.2) with s=1 reduces to the classical Schrödinger equations
(1.3)
-
ε
2
Δ
u
+
V
(
x
)
u
=
f
(
u
)
,
x
∈
ℝ
N
.
The existence, concentration and multiplicity of positive solutions for (1.3) have been extensively studied by many mathematicians. Rabinowitz [29] used the Mountain-Pass theorem to show that (1.3) possesses a positive ground state solution for ε>0 small provided that the potential V satisfies
V
∞
=
lim
¯
|
x
|
→
∞
V
(
x
)
>
V
0
=
inf
x
∈
ℝ
N
V
(
x
)
>
0
,
the nonlinearity f has subcritical growth and satisfies the well-known Ambrosetti–Rabinowitz condition ((AR) in short)
there exists μ>2 such that
μ
∫
0
s
f
(
τ
)
𝑑
τ
≤
f
(
s
)
s
for
s
>
0
,
and the monotonicity condition
f
(
s
)
/
s
is increasing for s>0.
The concentration behavior for the family of positive ground state solutions, which was obtained in [29], was proved by Wang [31]. Wang proved that the positive ground state solutions of (1.3) must concentrate around the global minima of V as ε→0. Under the same conditions on V(x) and f, Cingolani and Lazzo [12] proved the multiplicity of positive ground state solutions for (1.3) by using Ljusternik–Schnirelmann theory. Del Pino and Felmer [14] studied (1.3) with the conditions on V replaced by (V1) and (V2). They proved that (1.3) possesses a positive bound state solution for ε>0 small which concentrates around the local minima of V in Λ as ε→0. Moreover, we refer to [7, 8, 19, 27, 28, 33, 35].
In the nonlocal case, that is, when 0<s<1, there are only a few references on the existence and concentration phenomena for the singularly perturbed fractional Schrödinger equations. Dávila, del Pino and Wei [13] proved that if f(s)=sq-1(2<q<2s*), V(x)∈C1,α(ℝN)∩L∞(ℝN) and infℝNV(x)>0, then (1.1) has a multi-peak solution which concentrates around any finite subsets of the non-degenerate critical points of V. Alves and Miyagaki [1] used the penalization method due to del Pino and Felmer [14] to establish the existence of positive solutions of (1.1) which concentrate around the local minima of V provided that the potential V satisfies (V1) and (V2) and the nonlinearity f satisfies conditions (f1), (f4), (AR) and
there exists q∈(2,2s*) such that lim¯t→+∞f(t)/tq-1<+∞.
He and Zou [20] extended the results of Alves and Miyagaki [1] to the case where the nonlinearity f has critical growth.
Our main result is the following.
Theorem 1.2.
Assume that the potential V satisfies (V1) and (V2) and the nonlinearity f satisfies (f1)–(f3). If N≥4s, 2<q<2s*, or 2s<N<4s, 4s/(N-2s)<q<2s*, then there exists an ε0>0 such that for every λ>0, problem (1.1) possesses a positive bound state solution uε for all 0<ε<ε0. Moreover, the same conclusion holds provided that 2s<N<4s, 2<q≤4s/(N-2s) and λ>0 is sufficiently large. In addition, the following assertions hold:
There exists a maximum point
x
ε
of
u
ε
such that
lim
ε
→
0
dist
(
x
ε
,
ℳ
)
=
0
.
There exists
C
>
0
such that
u
ε
(
x
)
≤
C
ε
N
+
2
s
ε
N
+
2
s
+
|
x
-
x
ε
|
N
+
2
s
,
where
C
is independent of ε.
We note that, to the best of our knowledge, there is no result on the existence and concentration of positive bound state solutions for fractional Schrödinger equations under (f1)–(f3).
The proof of Theorem 1.2 is based on the variational method. The main difficulties lie in two aspects: First, the facts that the nonlinearity f(u) does not satisfy condition (AR) and that the function f(s)/s is not increasing for s>0 prevent us from obtaining a bounded Palais–Smale sequence ((PS) sequence in short) and using the Nehari manifold, respectively. The methods developed by Alves and Miyagaki [1] and He and Zou [20] cannot be applied to (1.1). Second, the unboundedness of the domain ℝN and the nonlinearity f(u) with critical growth lead to the lack of compactness. To complete this section, we sketch our proof.
To treat the nonlocal problem (1.1), we use the Caffarelli and Silvestre extension method [10] to study the corresponding extension problem
(1.4)
{
-
div
(
y
1
-
2
s
∇
w
)
=
0
in
ℝ
+
N
+
1
,
-
k
s
lim
y
→
0
+
y
1
-
2
s
∂
w
∂
y
(
x
,
y
)
=
-
V
(
ε
x
)
w
+
f
(
w
)
on
ℝ
N
×
{
0
}
,
where
lim
y
→
0
+
y
1
-
2
s
∂
w
∂
y
(
x
,
y
)
=
-
1
k
s
(
-
Δ
)
s
u
(
x
)
and
k
s
=
2
2
s
-
1
Γ
(
s
)
/
Γ
(
1
-
s
)
.
In the study of the singularly perturbed problem (1.4), the “limiting problem” which is given as
(1.5)
{
-
div
(
y
1
-
2
s
∇
w
)
=
0
in
ℝ
+
N
+
1
,
-
k
s
lim
y
→
0
+
y
1
-
2
s
∂
w
∂
y
(
x
,
y
)
=
-
m
w
+
f
(
w
)
on
ℝ
N
×
{
0
}
,
m
>
0
,
plays a crucial role. In [11], Chang and Wang showed that under the subcritical Berestycki–Lions-type conditions
lim
t
→
0
+
f
(
t
)
/
t
=
0
,
there exists p∈(1,2s*-1) such that lim¯t→+∞f(t)/tp<∞,
there exists T>0 such that (m/2)T2<F(t):=∫0Tf(t)𝑑t.
problem (1.5) has a ground state solution. By the lack of compactness of the embedding Hs(ℝN)↪L2s*(ℝN) for the critical nonlinearity f, the existence of ground state solutions of (1.4) becomes more complicated. Recently, He [22] obtained the existence of ground states of (1.4) with critical growth conditions (f1)–(f3).
In the present paper, condition (V2) is local and condition (AR) is no longer satisfied. We need to combine the penalization methods developed by Byeon and Wang [8] and del Pino and Felmer [14], which help us to overcome the obstacle caused by the non-compactness due to the unboundedness of the domain and the lack of condition (AR). To this end, we should modify the functional corresponding to (1.1), and it will be shown that this type of modification will force the concentration phenomena to occur inside Λ. We will adopt some ideas of Byeon and Jeanjean [7] to obtain a critical point wε of the modified functional. To verify that the critical point wε is indeed a solution of the original problem (1.4), we establish a uniform L∞-norm estimate of wε (independent of ε) by using the Brezis–Kato-type arguments for the fractional Laplacian explored by [16]. Our method is closely related to the work of Byeon and Jeanjean [7]. But for the critical problem (1.1), the method of Byeon and Jeanjean [7] can not be used directly and more careful analysis is needed.
This paper is organized as follows: In Section 2, we give some preliminary results. In Section 3, we prove the main result, Theorem 1.2.
Notations.
We use the following notations:
ℝ
+
N
+
1
=
{
(
x
,
t
)
∈
ℝ
N
+
1
:
t
>
0
}
.
B
R
+
:=
{
(
x
,
y
)
∈
ℝ
N
+
1
:
y
>
0
,
|
(
x
,
y
)
|
<
R
}
.
Γ
R
0
:=
{
(
x
,
0
)
∈
∂
ℝ
+
N
+
1
:
|
x
|
<
R
}
.
2 Preliminaries
In this section, we collect some preliminary results. Recall that Ds(ℝN), for s∈(0,1), is defined by the completion of C0∞(ℝN) with respect to the Gagliardo norm
∥
u
∥
D
s
(
ℝ
N
)
=
(
∫
ℝ
2
N
|
u
(
x
)
-
u
(
y
)
|
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
)
1
/
2
,
and the embedding Ds(ℝN)↪L2s*(ℝN) is continuous, that is,
∥
u
∥
L
2
s
*
(
ℝ
N
)
≤
C
(
N
,
s
)
∥
u
∥
D
s
(
ℝ
N
)
by [26, Theorem 1]. The fractional Sobolev space Hs(ℝN) is defined by
H
s
(
ℝ
N
)
=
{
u
∈
L
2
(
ℝ
N
)
:
∫
ℝ
2
N
|
u
(
x
)
-
u
(
y
)
|
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
}
endowed with the norm
∥
u
∥
H
s
(
ℝ
N
)
=
∥
u
∥
D
s
(
ℝ
N
)
+
∥
u
∥
L
2
(
ℝ
N
)
.
For N>2s and R>0, we see from [3, Lemma 2.1] and [15, Theorem 6.7] that
H
s
(
ℝ
N
)
is continuously embedded in
L
p
(
ℝ
N
)
for
p
∈
[
2
,
2
s
*
]
,
(2.1)
H
s
(
Γ
R
0
(
0
)
)
is continuously embedded in
L
p
(
Γ
R
0
(
0
)
)
for
p
∈
[
1
,
2
s
*
]
(2.2)
H
s
(
Γ
R
0
(
0
)
)
↪
L
p
(
Γ
R
0
(
0
)
)
is compact for
p
∈
[
1
,
2
s
*
)
.
The fractional Laplacian (-Δ)su can be represented (see [15, Lemma 3.2]) as
(
-
Δ
)
s
u
(
x
)
=
-
1
2
C
(
N
,
s
)
∫
ℝ
N
(
u
(
x
+
y
)
+
u
(
x
-
y
)
-
2
u
(
x
)
)
|
y
|
N
+
2
s
𝑑
y
for all
x
∈
ℝ
N
,
where
C
(
N
,
s
)
=
(
∫
ℝ
N
(
1
-
cos
ξ
1
)
|
ξ
|
N
+
2
s
𝑑
ξ
)
-
1
,
ξ
=
(
ξ
1
,
ξ
2
,
…
,
ξ
N
)
.
Also, we see from [15, Propositions 3.4 and 3.6] that
∥
(
-
Δ
)
s
/
2
u
∥
L
2
(
ℝ
N
)
2
=
∫
ℝ
N
|
ξ
|
2
s
|
u
^
|
2
𝑑
ξ
=
1
2
C
(
N
,
s
)
∫
ℝ
2
N
|
u
(
x
)
-
u
(
y
)
|
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
for all u∈Hs(ℝN).
An important feature of the operator (-Δ)s, 0<s<1, is its nonlocal character. A common approach to deal with this problem was proposed by Caffarelli and Silvestre [10], allowing to transform (1.1) into a local problem via the Dirichlet–Neumann map in the domain ℝ+N+1:={(x,t)∈ℝN+1:t>0}. For u∈Ds(ℝN), the solution w∈Xs(ℝ+N+1) of
{
-
div
(
y
1
-
2
s
∇
w
)
=
0
in
ℝ
+
N
+
1
,
w
=
u
on
ℝ
N
×
{
0
}
is called s-harmonic extension of u, denoted by w=Es(u). The s-harmonic extension and the fractional Laplacian have explicit expressions in terms of the Poisson and the Riesz kernels, respectively:
w
(
x
,
y
)
=
P
y
s
*
u
(
x
)
=
∫
ℝ
N
P
y
s
(
x
-
ξ
)
u
(
ξ
)
𝑑
ξ
,
where
P
y
s
(
x
)
:=
c
(
N
,
s
)
y
2
s
(
|
x
|
2
+
y
2
)
(
N
+
2
s
)
/
2
with a constant c(N,s) such that (see [23])
∫
ℝ
N
P
1
s
(
x
)
𝑑
x
=
1
.
Here, the space Xs(ℝ+N+1) is defined as the completion of C0∞(ℝ+N+1¯) under the norm
∥
w
∥
X
s
(
ℝ
+
N
+
1
)
:=
(
∫
ℝ
+
N
+
1
k
s
y
1
-
2
s
|
∇
w
|
2
𝑑
x
𝑑
y
)
1
/
2
.
Moreover, X0s(BR+(0)), for each R>0, is defined as the completion of C0∞(BR+(0)∪ΓR0(0)) under the norm
∥
w
∥
X
0
s
(
B
R
+
(
0
)
)
:=
(
∫
B
R
+
(
0
)
k
s
y
1
-
2
s
|
∇
w
|
2
𝑑
x
𝑑
y
)
1
/
2
.
By [4], the map Es(⋅) is an isometry between Ds(ℝN) and Xs(ℝ+N+1), i.e. for w=Es(u),
∥
u
∥
D
s
(
ℝ
N
)
=
∥
w
∥
X
s
(
ℝ
+
N
+
1
)
.
On the other hand, for a function w∈Xs(ℝ+N+1) we shall denote its trace on ℝN×{0} as u(x):=Tr(w)=w(x,0). This trace operator is also well defined and it satisfies
∥
u
∥
D
s
(
ℝ
N
)
≤
∥
w
∥
X
s
(
ℝ
+
N
+
1
)
.
For any r∈(1,+∞), denote by Lr(ℝ+N+1,ya) the weighted Lebesgue space endowed with the norm
∥
W
∥
L
r
(
ℝ
+
N
+
1
,
y
a
)
:=
(
∫
ℝ
+
N
+
1
y
a
|
W
|
r
𝑑
x
𝑑
y
)
1
/
2
.
We have the following lemma.
Lemma 2.1 ([16, Proposition 2.1.1 and Lemma 2.1.2]).
The following assertions hold:
Let
R
>
0
and let
𝒯
be a subset of
X
s
(
ℝ
+
N
+
1
)
such that
sup
W
∈
𝒯
∫
ℝ
+
N
+
1
y
1
-
2
s
|
∇
W
|
2
𝑑
x
𝑑
y
<
∞
.
Then
𝒯
is pre-compact in
L
2
(
B
𝑅
+
,
y
1
-
2
s
)
.
There exists a constant
S
^
>
0
such that for all
W
∈
X
s
(
ℝ
+
N
+
1
)
it holds that
(
∫
ℝ
+
N
+
1
y
1
-
2
s
|
W
|
2
γ
0
𝑑
x
𝑑
y
)
1
/
(
2
γ
0
)
≤
S
^
(
∫
ℝ
+
N
+
1
y
1
-
2
s
|
∇
W
|
2
𝑑
x
𝑑
y
)
1
/
2
,
where
γ
0
=
1
+
2
N
-
2
s
.
Lemma 2.2 ([4, Theorem 2.1]).
For every w∈Xs(R+N+1), it holds that
S
(
s
,
N
)
(
∫
ℝ
N
|
u
|
2
s
*
𝑑
x
)
2
/
2
s
*
≤
∫
ℝ
+
N
+
1
y
1
-
2
s
|
∇
w
|
2
𝑑
x
𝑑
y
,
where u=Tr(w). The best constant takes the exact value
S
(
s
,
N
)
=
2
π
s
Γ
(
1
-
s
)
Γ
(
(
N
+
2
s
)
/
2
)
Γ
(
N
/
2
)
2
s
/
N
Γ
(
s
)
Γ
(
(
N
-
2
s
)
/
2
)
Γ
(
N
)
2
s
/
N
,
and it is achieved when uδ takes the form
u
δ
(
x
)
=
δ
(
N
-
2
s
)
/
2
(
|
x
|
2
+
δ
2
)
-
(
N
-
2
s
)
/
2
for some δ>0 and wδ=Es(uδ).
For
w
∈
X
0
s
(
B
R
+
(
0
)
)
,
its extension by zero outside BR+(0) can be approximated by functions with compact support in ℝ+N+1¯. Thus, Lemma 2.2, together with Hölder’s inequality, gives a trace inequality for bounded domains.
Lemma 2.3.
For any 1≤r≤2s* and every w∈X0s(BR+(0)), it holds that
(
∫
Γ
R
0
(
0
)
|
u
|
r
𝑑
x
)
2
/
r
≤
C
(
r
,
s
,
N
,
R
)
∫
B
R
+
(
0
)
y
1
-
2
s
|
∇
w
|
2
𝑑
x
𝑑
y
,
where u=Tr(w).
Lemma 2.4.
Let N>2s and r>0. If {un}n=1∞ is bounded in Hs(RN) and if
sup
z
∈
ℝ
N
∫
Γ
r
0
(
z
)
|
u
n
|
2
s
*
𝑑
x
→
0
as
n
→
∞
,
then un→0 in L2s*(RN).
Proof.
By (2.1),
∥
u
n
∥
L
2
s
*
(
Γ
r
0
(
z
)
)
2
s
*
≤
C
∥
u
n
∥
H
s
(
Γ
r
0
(
z
)
)
2
s
*
.
Then, for any λ>0,
∥
u
n
∥
L
2
s
*
(
Γ
r
0
(
z
)
)
2
s
*
≤
C
λ
∥
u
n
∥
H
s
(
Γ
r
0
(
z
)
)
2
s
*
⋅
λ
⋅
∥
u
n
∥
L
2
s
*
(
Γ
r
0
(
z
)
)
2
s
*
(
1
-
λ
)
.
Choosing λ=2/2s* and covering ℝN with balls of radius r in such a way that each point of ℝN is contained in at most N+1 balls, then
∥
u
n
∥
L
2
s
*
(
ℝ
N
)
2
s
*
≤
C
λ
(
N
+
1
)
∥
u
n
∥
H
s
(
ℝ
N
)
2
⋅
sup
z
∈
ℝ
N
(
∫
Γ
r
0
(
z
)
|
u
n
|
2
s
*
)
1
-
λ
.
Under the assumptions of the lemma, un→0 in L2s*(ℝN). ∎
Definition 2.5.
We say that a sequence {wn}n=1∞ is tight in Xs(ℝ+N+1) if for every η>0 there exists ρ>0 such that
∫
ℝ
+
N
+
1
∖
B
ρ
+
(
0
)
y
1
-
2
s
|
∇
w
n
|
2
𝑑
x
𝑑
y
≤
η
for any
n
∈
ℕ
.
Proposition 2.6 ([16, Proposition 2.2.3], Concentration-Compactness Principle).
Let {wn}n=1∞ be a bounded tight sequence in Xs(R+N+1) such that wn⇀w in Xs(R+N+1). Let μ and ν be two nonnegative measures on R+N+1 and RN, respectively, such that
lim
n
→
∞
y
1
-
2
s
|
∇
w
n
|
2
=
μ
𝑎𝑛𝑑
lim
n
→
∞
|
w
n
(
x
,
0
)
|
2
s
*
=
ν
in the sense of measures. Then there exist an at most countable set J and three families {xj}j∈J⊂RN, {μj}j∈J and {νj}j∈J, μj,νj≥0 such that the following assertions hold:
ν
=
|
w
(
x
,
0
)
|
2
s
*
+
∑
j
∈
J
ν
j
δ
x
j
.
μ
≥
y
1
-
2
s
|
∇
w
|
2
+
∑
j
∈
J
μ
j
δ
(
x
j
,
0
)
.
μ
j
≥
S
(
s
,
N
)
ν
j
2
/
2
s
*
for all
j
∈
J
.
Lemma 2.7.
Let {wn}n=1∞ be a bounded sequence in Xs(R+N+1), N>2s, such that wn⇀0 in Xs(R+N+1). Suppose that there exist R>0 and γ>0 such that
(2.3)
∫
Γ
R
0
(
0
)
|
w
n
(
x
,
0
)
|
2
s
*
𝑑
x
≥
γ
>
0
.
Moreover, suppose that
(2.4)
|
〈
I
0
′
(
w
n
)
,
φ
〉
|
=
o
(
1
)
∥
φ
∥
X
s
(
ℝ
+
N
+
1
)
for all
φ
∈
X
0
s
(
B
R
′
+
(
0
)
)
,
where R′>R>0, o(1)→0 as n→∞, and
(2.5)
I
0
(
w
)
=
k
s
2
∫
ℝ
+
N
+
1
y
1
-
2
s
|
∇
w
|
2
𝑑
x
𝑑
y
-
1
2
s
*
∫
ℝ
N
|
w
(
x
,
0
)
|
2
s
*
𝑑
x
,
w
∈
X
s
(
ℝ
+
N
+
1
)
.
Then there exist a sequence of points {zn}n=1∞⊂ΓR0(0)¯ and a sequence of positive numbers {σn}n=1∞ such that
w
¯
n
(
x
,
y
)
:=
σ
n
(
N
-
2
s
)
/
2
w
n
(
σ
n
(
x
,
y
)
+
(
z
n
,
0
)
)
⇀
w
¯
in
X
s
(
ℝ
+
N
+
1
)
,
where w¯ is a nontrivial solution of
(2.6)
{
-
div
(
y
1
-
2
s
∇
w
)
=
0
in
ℝ
+
N
+
1
,
-
k
s
lim
𝑦
→
0
+
y
1
-
2
s
∂
w
∂
y
(
x
,
y
)
=
|
w
|
2
s
*
-
2
w
on
ℝ
N
×
{
0
}
.
Moreover, σn→0.
Proof.
Let ϕ0∈C∞(ℝ+) satisfying
(2.7)
{
ϕ
0
(
t
)
=
1
if
0
≤
t
≤
1
,
ϕ
0
(
t
)
=
0
if
t
≥
2
,
0
≤
ϕ
0
(
t
)
≤
1
,
|
ϕ
0
′
(
t
)
|
≤
C
and define ϕR(x,y):=ϕ0((|x|2+y2)1/2/R), y≥0.
Since {wn}n=1∞ is a sequence with wn⇀0 in Xs(ℝ+N+1), then by Proposition 2.6 we obtain an at most countable index set Γ and sequences {xj}j∈Γ⊂ℝN and {νj}j∈Γ⊂(0,∞) such that
|
ϕ
R
(
x
,
0
)
w
n
(
x
,
0
)
|
2
s
*
⇀
∑
j
∈
Γ
ν
j
δ
x
j
.
Then there is at least one j¯∈Γ such that xj¯∈ΓR0(0)¯ with νj¯>0. Otherwise, wn(x,0)→0 in L2s*(ΓR0(0)), which contradicts (2.3).
Inspired by [32], we define the concentration function
Q
n
(
r
)
=
sup
z
∈
Γ
R
0
(
0
)
¯
∫
Γ
r
0
(
z
)
|
w
n
(
x
,
0
)
|
2
s
*
𝑑
x
.
Fix a small τ∈(0,(ksS(s,N))N/2s) and choose σn=σn(τ)>0 and zn∈ΓR0(0)¯ such that
(2.8)
∫
Γ
σ
n
0
(
z
n
)
|
w
n
(
x
,
0
)
|
2
s
*
𝑑
x
=
Q
n
(
σ
n
)
=
τ
.
Set w¯n(x,y)=σn(N-2s)/2wn(σn(x,y)+(zn,0)). We see that
(2.9)
Q
~
n
(
r
)
:=
sup
z
∈
S
¯
n
∫
Γ
r
0
(
z
)
|
w
¯
n
(
x
,
0
)
|
2
s
*
𝑑
x
=
sup
z
∈
Γ
R
0
(
0
)
¯
∫
Γ
σ
n
r
0
(
z
)
|
w
n
(
x
,
0
)
|
2
s
*
𝑑
x
=
Q
n
(
σ
n
r
)
,
where S¯n:={x∈ℝN:σnx+zn∈ΓR0(0)¯}. Equations (2.8) and (2.9) imply that
(2.10)
Q
~
n
(
1
)
=
∫
Γ
1
0
(
0
)
|
w
¯
n
(
x
,
0
)
|
2
s
*
𝑑
x
=
∫
Γ
σ
n
0
(
z
n
)
|
w
n
(
x
,
0
)
|
2
s
*
𝑑
x
=
Q
n
(
σ
n
)
=
τ
.
Now, we prove that there is a small τ∈(0,(ksS(s,N))N/2s) such that σn(τ)→0 as n→∞. Otherwise, for any ε>0, there exists Mε>0 such that σn(ε)≥Mε after passing to a subsequence. Then
∫
Γ
M
ε
0
(
x
j
¯
)
|
ϕ
R
(
x
,
0
)
w
n
(
x
,
0
)
|
2
s
*
𝑑
x
≤
sup
z
∈
Γ
R
0
(
0
)
¯
∫
Γ
σ
n
(
ε
)
0
(
z
)
|
w
n
(
x
,
0
)
|
2
s
*
𝑑
x
=
Q
n
(
σ
n
(
ε
)
)
=
ε
.
In particular,
(2.11)
ν
j
¯
≤
∫
Γ
M
ε
0
(
x
j
¯
)
|
ϕ
R
(
x
,
0
)
w
n
(
x
,
0
)
|
2
s
*
𝑑
x
+
o
(
1
)
≤
ε
+
o
(
1
)
for all
ε
>
0
,
where o(1)→0 as n→∞. Letting n→∞ and ε→0 in (2.11), we see that νj¯≤0, which contradicts νj¯>0.
Since
∫
ℝ
+
N
+
1
y
1
-
2
s
|
∇
w
¯
n
|
2
𝑑
x
𝑑
y
=
∫
ℝ
+
N
+
1
y
1
-
2
s
|
∇
w
n
|
2
𝑑
x
𝑑
y
,
we see that, up to a subsequence, there exists w¯∈Xs(ℝ+N+1) such that
(2.12)
w
¯
n
⇀
w
¯
in
X
s
(
ℝ
+
N
+
1
)
.
For any φ∈C0∞(ℝ+N+1¯), set φ~n(x,y):=σn(2s-N)/2φ((x-zn,y)/σn). Note that σn→0 and zn∈ΓR0(0)¯ imply that φ~n(x,y)∈C0∞(BR′+(0)∪ΓR′0(0)) for n large. Then we get from (2.4) that
k
s
∫
ℝ
+
N
+
1
y
1
-
2
s
∇
w
¯
n
⋅
∇
φ
d
x
d
y
-
∫
ℝ
N
|
w
¯
n
(
x
,
0
)
|
2
s
*
-
2
w
¯
n
(
x
,
0
)
φ
(
x
,
0
)
𝑑
x
=
k
s
∫
ℝ
+
N
+
1
y
1
-
2
s
∇
w
n
⋅
∇
φ
~
n
d
x
d
y
-
∫
ℝ
N
|
w
n
(
x
,
0
)
|
2
s
*
-
2
w
n
(
x
,
0
)
φ
~
n
(
x
,
0
)
𝑑
x
(2.13)
=
o
(
1
)
∥
φ
∥
X
s
(
ℝ
+
N
+
1
)
+
o
(
1
)
,
where we have used the scaling invariance of the Xs-norm. Letting n→∞ in (2.13), we see that w¯ is a solution of (2.6).
Next, we claim that, up to a subsequence,
(2.14)
w
¯
n
(
x
,
0
)
→
w
¯
(
x
,
0
)
in
L
2
s
*
(
Γ
1
0
(
0
)
)
.
For a fixed d0>1, we set ϕd0(x,y):=ϕ0((|x|2+y2)1/2/d0). By (2.12), we may assume that |∇(ϕd0w¯n)|2⇀μ and |ϕd0(x,0)w¯n(x,0)|2s*⇀ν, where μ and ν are two bounded nonnegative measures on ℝ+N+1 and ℝN, respectively. By Proposition 2.6, we obtain an at most countable index set J and sequences {xj}j∈Γ⊂ℝN and {μj}j∈Γ,{νj}j∈Γ⊂(0,∞) such that
(2.15)
μ
≥
y
1
-
2
s
|
∇
(
ϕ
d
0
w
¯
)
|
2
+
∑
j
∈
J
μ
j
δ
(
x
j
,
0
)
,
ν
=
|
ϕ
d
0
(
x
,
0
)
w
¯
(
x
,
0
)
|
2
s
*
+
∑
j
∈
J
ν
j
δ
x
j
,
μ
j
≥
S
(
s
,
N
)
(
ν
i
)
2
/
2
s
*
.
To prove (2.14), it suffices to show that {xj}j∈J∩Γ10(0)¯=∅. Suppose that there is a xj0∈Γ10(0)¯ for some j0∈J. Define the function ϕρ(x,y):=ϕ0((|x-xj0|2+y2)1/2/ρ)(y≥0) and set ϕ~ρ,n(x,y)=ϕρ((x-zn,y)/σn), y≥0. By the facts that
z
n
∈
Γ
R
0
(
0
)
¯
,
x
j
0
∈
Γ
1
0
(
0
)
¯
,
σ
n
→
0
as
n
→
∞
,
we see that for n large,
supp
ϕ
~
ρ
,
n
⊂
B
2
σ
n
ρ
+
(
z
n
+
σ
n
x
j
0
,
0
)
∪
Γ
2
σ
n
ρ
0
(
z
n
+
σ
n
x
j
0
)
⊂
(
B
R
′
+
(
0
)
∪
Γ
R
′
0
(
0
)
)
.
Then ϕ~ρ,nwn∈X0s(BR′+(0)). Since {wn} is bounded in Xs(ℝ+N+1), Hölder’s inequality, polar coordinate transformation and Lemma 2.1 (ii) yield
∫
ℝ
+
N
+
1
y
1
-
2
s
|
∇
(
ϕ
~
ρ
,
n
w
n
)
|
2
𝑑
x
𝑑
y
≤
C
∫
ℝ
+
N
+
1
y
1
-
2
s
|
∇
ϕ
~
ρ
,
n
|
2
w
n
2
𝑑
x
𝑑
y
+
C
∫
ℝ
+
N
+
1
y
1
-
2
s
ϕ
~
ρ
,
n
2
|
∇
w
n
|
2
𝑑
x
𝑑
y
≤
C
(
∫
ℝ
+
N
+
1
y
1
-
2
s
|
w
n
|
2
γ
0
𝑑
x
𝑑
y
)
1
/
γ
0
(
∫
ℝ
+
N
+
1
y
1
-
2
s
|
∇
ϕ
~
ρ
,
n
|
N
+
2
-
2
s
𝑑
x
𝑑
y
)
2
/
(
N
+
2
-
2
s
)
+
C
≤
C
(
σ
n
ρ
)
2
(
∫
B
2
σ
n
ρ
+
(
y
n
+
σ
n
x
j
0
,
0
)
y
1
-
2
s
𝑑
x
𝑑
y
)
2
/
(
N
+
2
-
2
s
)
+
C
(2.16)
≤
C
(
σ
n
ρ
)
2
(
∫
0
2
σ
n
ρ
r
N
+
1
-
2
s
𝑑
r
)
2
/
(
N
+
2
-
2
s
)
+
C
≤
C
.
Hence {ϕ~ρ,nwn} is bounded in Xs(ℝ+N+1) and the bound is independent of ρ. By (2.4), we get
k
s
∫
ℝ
+
N
+
1
y
1
-
2
s
∇
w
¯
n
⋅
∇
(
ϕ
ρ
w
¯
n
)
d
x
d
y
-
∫
ℝ
N
|
w
¯
n
(
x
,
0
)
|
2
s
*
-
2
w
¯
n
(
x
,
0
)
(
ϕ
ρ
(
x
,
0
)
w
¯
n
(
x
,
0
)
)
𝑑
x
=
k
s
∫
ℝ
+
N
+
1
y
1
-
2
s
∇
w
n
⋅
∇
(
ϕ
~
ρ
,
n
w
n
)
d
x
d
y
-
∫
ℝ
N
|
w
n
(
x
,
0
)
|
2
s
*
-
2
w
n
(
x
,
0
)
(
ϕ
~
ρ
,
n
(
x
,
0
)
w
n
(
x
,
0
)
)
𝑑
x
(2.17)
=
o
(
1
)
∥
ϕ
~
ρ
,
n
w
n
∥
X
s
(
ℝ
+
N
+
1
)
=
o
(
1
)
.
As ρ→0, by Lemma 2.1 (i) and (ii) and similar to (2.16), we have
lim
¯
n
→
∞
|
∫
ℝ
+
N
+
1
y
1
-
2
s
(
∇
w
¯
n
⋅
∇
ϕ
ρ
)
w
¯
n
𝑑
x
𝑑
y
|
≤
lim
¯
n
→
∞
(
∫
ℝ
+
N
+
1
y
1
-
2
s
|
∇
w
¯
n
|
2
𝑑
x
𝑑
y
)
1
/
2
⋅
(
∫
B
2
ρ
+
(
x
j
0
,
0
)
y
1
-
2
s
|
w
¯
n
|
2
|
∇
ϕ
ρ
|
2
𝑑
x
𝑑
y
)
1
/
2
≤
C
(
∫
B
2
ρ
+
(
x
j
0
,
0
)
y
1
-
2
s
|
w
¯
|
2
|
∇
ϕ
ρ
|
2
𝑑
x
𝑑
y
)
1
/
2
≤
C
(
∫
B
2
ρ
+
(
x
j
0
,
0
)
y
1
-
2
s
|
w
¯
|
2
γ
0
𝑑
x
𝑑
y
)
1
/
2
γ
0
⋅
(
∫
B
2
ρ
+
(
x
j
0
,
0
)
y
1
-
2
s
|
∇
ϕ
ρ
|
N
+
2
-
2
s
𝑑
x
𝑑
y
)
1
/
(
N
+
2
-
2
s
)
≤
C
(
∫
B
2
ρ
+
(
x
j
0
,
0
)
y
1
-
2
s
|
w
¯
|
2
γ
0
𝑑
x
𝑑
y
)
1
/
2
γ
0
→
0
,
and further
lim
¯
n
→
∞
∫
ℝ
+
N
+
1
y
1
-
2
s
|
∇
w
¯
n
|
2
ϕ
ρ
𝑑
x
𝑑
y
≥
∫
ℝ
+
N
+
1
y
1
-
2
s
|
∇
w
¯
|
2
ϕ
ρ
𝑑
x
𝑑
y
+
μ
j
0
→
μ
j
0
and
lim
¯
n
→
∞
∫
ℝ
N
|
w
¯
n
(
x
,
0
)
|
2
s
*
ϕ
ρ
(
x
,
0
)
𝑑
x
=
∫
ℝ
N
|
w
¯
(
x
,
0
)
|
2
s
*
ϕ
ρ
(
x
,
0
)
𝑑
x
+
ν
j
0
→
ν
j
0
,
where we have used the facts that for ρ>0 small, ϕd0w¯n=w¯n on
supp
ϕ
ρ
⊂
(
B
2
ρ
+
(
x
j
0
,
0
)
∪
Γ
2
ρ
0
(
x
j
0
,
0
)
)
⊂
(
B
d
0
+
(
0
)
∪
Γ
d
0
0
(
0
)
)
.
We see from (2.17) that ksμj0≤νj0. Then by (2.15) we get νj0≥(ksS(s,N))N/2s. By (2.10),
(2.18)
(
k
s
S
(
s
,
N
)
)
N
/
2
s
≤
ν
j
0
≤
∫
Γ
1
0
(
0
)
|
w
¯
n
(
x
,
0
)
|
2
s
*
𝑑
x
+
o
(
1
)
=
τ
+
o
(
1
)
,
where o(1)→0 as n→∞. Letting n→∞ in (2.18), we see that (ksS(s,N))N/2s≤τ which contradicts τ<(ksS(s,N))N/2s. Hence,
{
x
j
}
j
∈
J
∩
Γ
1
0
(
0
)
¯
=
∅
,
and (2.14) holds. Equation (2.10) and (2.14) yield
∫
Γ
1
0
(
0
)
|
w
¯
(
x
,
0
)
|
2
s
*
𝑑
x
=
lim
n
→
∞
∫
Γ
1
0
(
0
)
|
w
¯
n
(
x
,
0
)
|
2
s
*
𝑑
x
=
τ
>
0
,
which means that w¯ is nontrivial. ∎
The corresponding energy functional to the “limiting equation” (1.5) is
I
m
(
w
)
=
k
s
2
∫
ℝ
+
N
+
1
y
1
-
2
s
|
∇
w
|
2
𝑑
x
𝑑
y
+
m
2
∫
ℝ
N
w
2
(
x
,
0
)
𝑑
x
-
∫
ℝ
N
F
(
w
(
x
,
0
)
)
𝑑
x
,
w
∈
X
1
,
s
(
ℝ
+
N
+
1
)
,
where
X
1
,
s
(
ℝ
+
N
+
1
)
:=
{
w
∈
X
s
(
ℝ
+
N
+
1
)
:
∫
ℝ
N
w
2
(
x
,
0
)
𝑑
x
<
∞
}
equipped with the norm
∥
w
∥
X
1
,
s
(
ℝ
+
N
+
1
)
=
(
∫
ℝ
+
N
+
1
y
1
-
2
s
|
∇
w
|
2
𝑑
x
𝑑
y
+
∫
ℝ
N
w
2
(
x
,
0
)
𝑑
x
)
1
/
2
.
In view of [11, 30], if w∈X1,s(ℝ+N+1) is a weak solution to (1.5), the following Pohozaev’s identity holds:
P
m
(
w
)
=
k
s
(
N
-
2
s
)
2
∫
ℝ
+
N
+
1
y
1
-
2
s
|
∇
w
|
2
𝑑
x
𝑑
y
+
m
N
2
∫
ℝ
N
w
2
(
x
,
0
)
𝑑
x
-
N
∫
ℝ
N
F
(
w
(
x
,
0
)
)
𝑑
x
=
0
.
By [22] (see also [2, 17]), we have the following proposition.
Proposition 2.8.
If N≥4s, 2<q<2s*, or 2s<N<4s, 4s/(N-2s)<q<2s*, then for all λ>0, problem (1.5) possesses a positive ground state solution with cm<sN(ksS(s,N))N/(2s), where
c
m
:=
inf
γ
∈
Γ
m
sup
t
∈
[
0
,
1
]
I
m
(
γ
(
t
)
)
is the Mountain-Pass level, which is also the ground state energy of (1.5). Moreover, if 2s<N<4s and 2<q≤4s/(N-2s), then for λ>0 sufficiently large the same conclusion holds.
Let Sm be the set of ground state solutions W of (1.5) satisfying W(0,0)=maxx∈ℝNW(x,0). Then we obtain the following compactness of Sm.
Proposition 2.9.
For each m>0, the set Sm is compact in X1,s(R+N+1).
Proof.
Arguing as in the proof of Proposition 2.8 (see [22]), we verify that {Wk}k=1∞ is bounded in X1,s(ℝ+N+1) for any {Wk}k=1∞⊂Sm, and up to a subsequence there exist {xk}k=1∞⊂ℝN and W¯(x,y)∈X1,s(ℝ+N+1) such that W¯k(x,y):=Wk(x+xk,y)⇀W¯(x,y) in X1,s(ℝ+N+1) and W¯k(x,y)→W¯(x,y) in Xs(ℝ+N+1) as k→∞. Then by Lemma 2.2,
(2.19)
W
¯
k
(
x
,
0
)
→
W
¯
(
x
,
0
)
in
L
2
s
*
(
ℝ
N
)
.
By (2.19), for any fixed δ>0, there exists R=R(δ)>0 such that
∫
{
W
¯
k
(
x
,
0
)
>
R
}
|
W
¯
k
(
x
,
0
)
|
2
s
*
𝑑
x
<
δ
uniformly for k∈ℕ. By adapting the Brezis–Kato-type arguments explored in [16, Proposition 4.1.1], we see that there exists C0>0 such that for all k∈ℕ,
(2.20)
∥
W
¯
k
(
x
,
0
)
∥
L
∞
(
ℝ
N
)
=
∥
W
k
(
x
,
0
)
∥
L
∞
(
ℝ
N
)
≤
C
0
.
Next, we will show that
(2.21)
lim
|
x
|
→
+
∞
W
¯
k
(
x
,
0
)
=
0
uniformly for
k
∈
ℕ
.
Indeed, since W¯k(x,y) is a solution of (1.5), obviously W¯k(x,0)∈Hs(ℝN) satisfies
(
-
Δ
)
s
u
+
m
u
=
f
(
u
)
.
Combining (f1), (f2), (2.19) and (2.20), we have
(2.22)
f
(
W
¯
k
(
x
,
0
)
)
→
f
(
W
¯
(
x
,
0
)
)
in
L
p
(
ℝ
N
)
,
p
∈
[
2
,
+
∞
)
,
and
(2.23)
∥
f
(
W
¯
k
(
x
,
0
)
)
∥
L
∞
(
ℝ
N
)
≤
C
for all
n
∈
ℕ
.
Using the Fourier transform, we see that
W
¯
k
(
x
,
0
)
=
∫
ℝ
N
𝒦
m
(
x
-
z
)
f
(
W
¯
k
(
z
,
0
)
)
𝑑
z
,
where
𝒦
m
(
x
)
=
ℱ
-
1
(
1
|
ξ
|
2
s
+
m
)
is the Bessel Kernel. In view of [18, Theorem 3.3], we see that 𝒦m satisfies the following conditions:
𝒦
m
is positive, radially symmetric and smooth in ℝN∖{0}.
There exists C>0 such that
𝒦
m
(
x
)
≤
C
|
x
|
N
+
2
s
for all
x
∈
ℝ
N
∖
{
0
}
.
𝒦
m
∈
L
q
(
ℝ
N
)
for all q∈[1,2s*/2).
For any δ>0 small, we have
0
≤
W
¯
k
(
x
,
0
)
=
∫
ℝ
N
∖
Γ
1
/
δ
0
(
x
)
𝒦
m
(
x
-
z
)
f
(
W
¯
k
(
z
,
0
)
)
𝑑
z
+
∫
Γ
1
/
δ
0
(
x
)
𝒦
m
(
x
-
z
)
f
(
W
¯
k
(
z
,
0
)
)
𝑑
z
(2.24)
=
(I)
+
(II)
.
By (K1), (K2) and (2.23), we see that
(2.25)
(I)
≤
C
δ
s
∫
ℝ
N
∖
Γ
1
/
δ
0
(
x
)
1
|
x
-
z
|
N
+
s
𝑑
z
≤
C
δ
s
∫
ℝ
N
∖
Γ
1
0
(
0
)
1
|
z
|
N
+
s
𝑑
z
≤
C
δ
s
.
On the other hand, we choose q>1 which is close to 1 and q′>2 such that 1q+1q′=1. Then by (K3) we obtain
(II)
≤
∫
Γ
1
/
δ
0
(
x
)
𝒦
m
(
x
-
z
)
|
f
(
W
¯
k
(
z
,
0
)
)
-
f
(
W
¯
(
z
,
0
)
)
|
𝑑
z
+
∫
Γ
1
/
δ
0
(
x
)
𝒦
m
(
x
-
z
)
f
(
W
¯
(
z
,
0
)
)
𝑑
z
≤
∥
𝒦
m
∥
L
q
(
ℝ
N
)
(
∥
f
(
W
¯
k
(
z
,
0
)
)
-
f
(
W
¯
(
z
,
0
)
)
∥
L
q
′
(
ℝ
N
)
+
∥
f
(
W
¯
(
z
,
0
)
)
∥
L
q
′
(
Γ
1
/
δ
0
(
x
)
)
)
By (2.22),
∥
f
(
W
¯
k
(
z
,
0
)
)
-
f
(
W
¯
(
z
,
0
)
)
∥
L
q
′
(
ℝ
N
)
→
0
as
k
→
∞
,
and
∥
f
(
W
¯
(
z
,
0
)
)
∥
L
q
′
(
Γ
1
/
δ
0
(
x
)
)
→
0
as
|
x
|
→
+
∞
,
we deduce that for the above δ>0 there exists R0>0 and k0∈ℕ such that for any k≥k0 and |x|≥R0 we have (II)≤δ. The same approach can be used to show that for each k<k0 there exists Rk>0 such that for |x|≥Rk we have (II)≤δ. Hence, letting R=max{R0,R1,…,Rk0-1} yields
(2.26)
(II)
≤
δ
,
|
x
|
≥
R
.
Inequalities (2.24), (2.25), (2.26) and the arbitrariness of δ give (2.21). We see from (f1) and (f2) that
k
s
∫
ℝ
+
N
+
1
y
1
-
2
s
|
∇
W
k
|
2
𝑑
x
𝑑
y
+
m
∫
ℝ
N
W
k
2
(
x
,
0
)
𝑑
x
=
∫
ℝ
N
f
(
W
k
(
x
,
0
)
)
W
k
(
x
,
0
)
𝑑
x
≤
m
2
∫
ℝ
N
W
k
2
(
x
,
0
)
𝑑
x
+
C
∫
ℝ
N
W
k
2
s
*
(
x
,
0
)
𝑑
x
.
Then
(2.27)
∫
ℝ
N
W
k
2
(
x
,
0
)
𝑑
x
≤
C
∫
ℝ
N
W
k
2
s
*
(
x
,
0
)
𝑑
x
≤
C
W
k
2
s
*
-
2
(
0
,
0
)
∫
ℝ
N
W
k
2
(
x
,
0
)
𝑑
x
,
which implies that W¯k(-xk,0)=Wk(0,0)≥C>0 for k∈ℕ. Then, by (2.21), {xk}k=1∞ must be bounded, i.e. there exists r0>0, such that |xk|≤r0. By (2.19), we see that for any δ>0 there exists Rδ>0 such that for all k∈ℕ,
(2.28)
∫
ℝ
N
∖
Γ
R
δ
+
r
0
0
(
0
)
W
k
2
s
*
(
x
,
0
)
𝑑
x
≤
∫
ℝ
N
∖
Γ
R
δ
0
(
0
)
W
¯
k
2
s
*
(
x
,
0
)
𝑑
x
<
δ
.
On the other hand, since {Wk}k=1∞ is bounded in X1,s(ℝ+N+1), we may assume that there exists W∈X1,s(ℝ+N+1) such that
W
k
(
x
,
y
)
⇀
W
(
x
,
y
)
in
X
1
,
s
(
ℝ
+
N
+
1
)
,
W
k
(
x
,
0
)
→
W
(
x
,
0
)
in
L
loc
p
(
ℝ
N
)
,
1
≤
p
<
2
s
*
,
W
k
(
x
,
0
)
→
W
(
x
,
0
)
a.e. on
ℝ
N
.
We choose a small δ0>0 and see from (2.20) that, as k→∞,
(2.29)
∫
Γ
R
δ
+
r
0
0
(
0
)
|
W
k
(
x
,
0
)
-
W
(
x
,
0
)
|
2
s
*
𝑑
x
≤
C
∫
Γ
R
δ
+
r
0
0
(
0
)
|
W
k
(
x
,
0
)
-
W
(
x
,
0
)
|
2
s
*
-
δ
0
𝑑
x
→
0
.
Inequalities (2.28) and (2.29) and the Interpolation Inequality show that
W
k
(
x
,
0
)
→
W
(
x
,
0
)
in
L
p
(
ℝ
N
)
,
2
<
p
≤
2
s
*
.
Combining the fact 〈Im′(Wk)-Im′(W),Wk-W〉=0 with (f1) and (f2), we conclude that Wk→W in X1,s(ℝ+N+1). This completes the proof that Sm is compact in X1,s(ℝ+N+1). ∎
3 Proof of Theorem 1.2
In view of [10], we study problem (1.4), which is equivalent to (1.1). We are looking for positive solutions in the Hilbert space Xε defined by
X
ε
=
{
w
∈
X
1
,
s
(
ℝ
+
N
+
1
)
:
∫
ℝ
N
V
(
ε
x
)
w
2
(
x
,
0
)
𝑑
x
<
∞
}
endowed with the norm
∥
w
∥
X
ε
=
(
∫
ℝ
+
N
+
1
k
s
y
1
-
2
s
|
∇
w
|
2
𝑑
x
𝑑
y
+
∫
ℝ
N
V
(
ε
x
)
w
2
(
x
,
0
)
𝑑
x
)
1
/
2
.
Note that if the function wε∈Xε is a solution of (1.4), then the function uε(x)=Tr(wε)=wε(x,0) is a solution of (1.1).
Inspired by [8, 14], we define the truncated function
g
ε
(
x
,
t
)
:=
χ
Λ
/
ε
(
x
)
f
(
t
)
+
(
1
-
χ
Λ
/
ε
(
x
)
)
min
{
κ
0
α
t
,
f
(
t
)
}
of f(t), and set
G
ε
(
x
,
t
)
:=
∫
0
t
g
ε
(
x
,
τ
)
𝑑
τ
,
where χΛ/ε is the characteristic function of Λ/ε and 0<κ0<1.
Finally, set
χ
ε
(
x
)
=
{
0
if
x
∈
Λ
/
ε
,
ε
-
1
if
x
∉
Λ
/
ε
and define an auxiliary functional Jε:Xε→ℝ given by
J
ε
(
w
)
:=
k
s
2
∫
ℝ
+
N
+
1
y
1
-
2
s
|
∇
w
|
2
𝑑
x
𝑑
y
+
1
2
∫
ℝ
N
V
(
ε
x
)
w
2
(
x
,
0
)
𝑑
x
-
∫
ℝ
N
G
ε
(
x
,
w
(
x
,
0
)
)
𝑑
x
+
Q
ε
(
w
)
,
where
Q
ε
(
w
)
:=
(
∫
ℝ
N
χ
ε
(
x
)
w
2
(
x
,
0
)
𝑑
x
-
1
)
+
2
.
As we will see later, this type of penalization will force the concentration phenomena to occur inside Λ, and it is standard to show that Jε∈C1(Xε,ℝ).
Let cV0=IV0(W) for W∈SV0 and 10δ=dist{ℳ,ℝN∖Λ}. We fix a β∈(0,δ) and introduce a cut-off function ϕ0∈C∞(ℝ+) which satisfies (2.7). We will find a solution of (1.4) near the set
E
ε
:=
{
ϕ
0
(
(
|
ε
x
-
x
′
|
2
+
ε
2
y
2
)
1
/
2
/
β
)
W
(
x
-
(
x
′
/
ε
)
,
y
)
:
x
′
∈
ℳ
β
,
W
∈
S
V
0
}
for ε>0 small, where ℳβ:={y∈ℝN:infz∈ℳ|y-z|≤β}. Now, we define
J
ε
α
:=
{
w
∈
X
ε
:
J
ε
(
w
)
≤
α
}
,
and for A⊂Xε we use the notation
A
a
:=
{
w
∈
X
ε
:
inf
v
∈
A
∥
w
-
v
∥
X
ε
≤
a
}
.
Fix W*∈SV0 and define Wε,t(x,y):=ϕ0(ε(|x|2+y2)1/2/β)W*(x/t,y/t). Next, we will show that Jε possesses the Mountain-Pass geometry.
Direct computations show that
I
V
0
(
W
*
(
x
/
t
,
y
/
t
)
)
=
I
V
0
(
W
*
(
x
/
t
,
y
/
t
)
)
-
1
N
t
N
P
m
(
W
*
)
=
(
1
2
t
N
-
2
s
-
N
-
2
s
2
N
t
N
)
∥
W
*
∥
X
s
(
ℝ
+
N
+
1
)
2
.
Taking a large t0>0 such that IV0(W*(x/t0,y/t0))<-3, by Hölder’s inequality, polar coordinates transformation and Lemma 2.1 (ii) we see that
∫
ℝ
+
N
+
1
y
1
-
2
s
|
∇
ϕ
0
(
ε
(
|
x
|
2
+
y
2
)
1
/
2
/
β
)
|
2
(
W
*
(
x
/
t
0
,
y
/
t
0
)
)
2
𝑑
x
𝑑
y
≤
C
ε
2
∫
B
2
β
/
ε
+
(
0
)
∖
B
β
/
ε
+
(
0
)
y
1
-
2
s
(
W
*
(
x
/
t
0
,
y
/
t
0
)
)
2
𝑑
x
𝑑
y
≤
C
ε
2
[
∫
B
2
β
/
ε
+
(
0
)
∖
B
β
/
ε
+
(
0
)
y
1
-
2
s
(
W
*
(
x
/
t
0
,
y
/
t
0
)
)
2
γ
0
𝑑
x
𝑑
y
]
1
/
γ
0
[
∫
B
2
β
/
ε
+
(
0
)
∖
B
β
/
ε
+
(
0
)
y
1
-
2
s
𝑑
x
𝑑
y
]
1
-
(
1
/
γ
0
)
≤
C
ε
2
[
∫
B
2
β
/
ε
+
(
0
)
∖
B
β
/
ε
+
(
0
)
y
1
-
2
s
(
W
*
(
x
/
t
0
,
y
/
t
0
)
)
2
γ
0
𝑑
x
𝑑
y
]
1
/
γ
0
[
∫
β
/
ε
2
β
/
ε
r
N
+
1
-
2
s
𝑑
r
]
1
-
(
1
/
γ
0
)
(3.1)
≤
C
[
∫
B
2
β
/
ε
+
(
0
)
∖
B
β
/
ε
+
(
0
)
y
1
-
2
s
(
W
*
(
x
/
t
0
,
y
/
t
0
)
)
2
γ
0
𝑑
x
𝑑
y
]
1
/
γ
0
→
0
as
ε
→
0
.
Note that 0∈ℳ. Then Qε(Wε,t0)=0 and gε(x,Wε,t0(x,0))=f(Wε,t0(x,0)). Direct computations and (3.1) show that
J
ε
(
W
ε
,
t
0
)
=
I
ε
(
W
ε
,
t
0
)
=
I
V
0
(
W
*
(
x
/
t
0
,
y
/
t
0
)
)
+
o
(
1
)
<
-
2
for
ε
>
0
small.
By (f1), (f2) and Lemma 2.2, we see that Jε(w)>0 for ∥w∥Xε small. Hence, we can define the Mountain-Pass value of Jε by
c
ε
:=
inf
γ
∈
Γ
ε
max
s
∈
[
0
,
1
]
J
ε
(
γ
(
s
)
)
,
where Γε:={γ∈C([0,1],Xε):γ(0)=0,γ(1)=Wε,t0}.
Let γε(t′)=Wε,t′t0 for t′∈(0,1] and γε(0)=0. Set c~ε:=maxt′∈[0,1]Jε(γε(t′)). Then, as in [7, Propositions 2 and 3], we can prove the following proposition.
Proposition 3.1.
One has cε≤c~ε and limε→0cε=limε→0c~ε=cV0.
We have the following lemma and this lemma is a key for the proof of Theorem 1.2.
Lemma 3.2.
There exists a d0>0 such that for any {εi}i=1∞ and {wεi}i=1∞ with
lim
i
→
∞
ε
i
=
0
,
w
ε
i
∈
E
ε
i
d
0
,
lim
i
→
∞
J
ε
i
(
w
ε
i
)
≤
c
V
0
,
lim
i
→
∞
∥
J
ε
i
′
(
w
ε
i
)
∥
(
X
ε
i
)
-
1
=
0
,
there exists, up to a subsequence, {zi}i=1∞⊂RN, x0∈M and W∈SV0 such that
lim
i
→
∞
|
ε
i
z
i
-
x
0
|
=
0
𝑎𝑛𝑑
lim
i
→
∞
∥
w
ε
i
-
ϕ
0
(
ε
i
(
|
x
-
z
i
|
2
+
y
2
)
1
/
2
/
β
)
W
(
x
-
z
i
,
y
)
∥
X
ε
i
=
0
.
Proof.
For notational simplicity, we write ε for εi and still use ε after taking a subsequence. By the definition of Eεd0 and the compactness of SV0 and ℳβ, we see that there exist W0∈SV0 and {xε}⊂ℳβ such that for ε>0 small,
(3.2)
∥
w
ε
-
ϕ
0
(
ε
(
|
x
-
(
x
ε
/
ε
)
|
2
+
y
2
)
1
/
2
/
β
)
W
0
(
x
-
(
x
ε
/
ε
)
,
y
)
∥
X
ε
≤
2
d
0
,
and, as ε→0,
x
ε
→
x
0
∈
ℳ
β
.
Next, we claim that there exist {wε,1},{wε,2}⊂Xε and {kε},{jε}⊂ℕ such that the following conditions hold:
k
ε
≤
β
/
(
5
ε
)
and kε→∞ as ε→0, 0≤jε≤kε-1 and |wε,i|≤|wε|, i=1,2.
One has
w
ε
,
1
=
w
ε
in
B
(
2
β
/
ε
)
+
(
5
j
ε
+
1
)
k
ε
+
(
x
ε
/
ε
,
0
)
,
w
ε
,
2
=
w
ε
in
ℝ
+
N
+
1
∖
B
(
2
β
/
ε
)
+
(
5
j
ε
+
4
)
k
ε
+
(
x
ε
/
ε
,
0
)
.
One has
supp
w
ε
,
1
⊂
B
(
2
β
/
ε
)
+
(
5
j
ε
+
2
)
k
ε
+
(
x
ε
/
ε
,
0
)
¯
and
supp
w
ε
,
2
⊂
ℝ
+
N
+
1
∖
B
(
2
β
/
ε
)
+
(
5
j
ε
+
3
)
k
ε
+
(
x
ε
/
ε
,
0
)
.
∥
w
ε
-
w
ε
,
1
-
w
ε
,
2
∥
X
ε
→
0
as ε→0.
∥
w
ε
∥
X
s
(
B
j
ε
,
ε
)
→
0
and
∫
B
j
ε
,
ε
y
1
-
2
s
|
w
ε
|
2
γ
0
𝑑
x
𝑑
y
→
0
as
ε
→
0
,
where
B
j
ε
,
ε
:=
B
(
2
β
/
ε
)
+
5
(
j
ε
+
1
)
k
ε
+
(
x
ε
/
ε
,
0
)
¯
∖
B
(
2
β
/
ε
)
+
5
j
ε
k
ε
+
(
x
ε
/
ε
,
0
)
.
∥
w
ε
(
x
,
0
)
∥
L
2
s
*
(
Γ
j
ε
,
ε
)
→
0
and
∫
Γ
j
ε
,
ε
V
(
ε
x
)
|
w
ε
(
x
,
0
)
|
2
𝑑
x
→
0
as
ε
→
0
,
where
Γ
j
ε
,
ε
:=
Γ
(
2
β
/
ε
)
+
5
(
j
ε
+
1
)
k
ε
0
(
x
ε
/
ε
)
¯
∖
Γ
(
2
β
/
ε
)
+
5
j
ε
k
ε
0
(
x
ε
/
ε
)
.
Letting kε∈ℕ such that kε≤β/(5ε) and kε→∞ as ε→0, and setting w~ε(x,y):=wε(x+(xε/ε),y), we see from (3.2), Lemma 2.1 (ii), Lemma 2.2 and the fact ϕ0(ε(|x|2+y2)/β)≡0 in ℝ+N+1∖B2β/ε+(0) that
∫
ℝ
+
N
+
1
∖
B
2
β
/
ε
+
(
0
)
y
1
-
2
s
|
∇
w
~
ε
|
2
𝑑
x
𝑑
y
+
∫
ℝ
N
∖
Γ
2
β
/
ε
0
(
0
)
V
(
ε
x
+
x
ε
)
|
w
~
ε
(
x
,
0
)
|
2
𝑑
x
(3.3)
+
∫
ℝ
+
N
+
1
∖
B
2
β
/
ε
+
(
0
)
y
1
-
2
s
|
w
~
ε
|
2
γ
0
𝑑
x
𝑑
y
+
∫
ℝ
N
∖
Γ
2
β
/
ε
0
(
0
)
|
w
~
ε
(
x
,
0
)
|
2
s
*
𝑑
x
≤
C
d
0
.
Denoting
B
~
j
,
ε
:=
B
(
2
β
/
ε
)
+
5
(
j
+
1
)
k
ε
+
(
0
)
¯
∖
B
(
2
β
/
ε
)
+
5
j
k
ε
+
(
0
)
and
Γ
~
j
,
ε
:=
Γ
(
2
β
/
ε
)
+
5
(
j
+
1
)
k
ε
0
(
0
)
¯
∖
Γ
(
2
β
/
ε
)
+
5
j
k
ε
0
(
0
)
for j=0,1,…,kε-1, we see from (3.3) that
∑
j
=
0
k
ε
-
1
∫
B
~
j
,
ε
y
1
-
2
s
|
∇
w
~
ε
|
2
𝑑
x
𝑑
y
+
∑
j
=
0
k
ε
-
1
∫
Γ
~
j
,
ε
V
(
ε
x
+
x
ε
)
|
w
~
ε
(
x
,
0
)
|
2
𝑑
x
+
∑
j
=
0
k
ε
-
1
∫
B
~
j
,
ε
y
1
-
2
s
|
w
~
ε
|
2
γ
0
𝑑
x
𝑑
y
+
∑
j
=
0
k
ε
-
1
∫
Γ
~
j
,
ε
|
w
~
ε
(
x
,
0
)
|
2
s
*
𝑑
x
≤
C
d
0
.
Thus, there exists a jε∈{0,1,…,kε-1} such that
∫
B
~
j
ε
,
ε
y
1
-
2
s
|
∇
w
~
ε
|
2
𝑑
x
𝑑
y
+
∫
Γ
~
j
ε
,
ε
V
(
ε
x
+
x
ε
)
|
w
~
ε
(
x
,
0
)
|
2
𝑑
x
(3.4)
+
∫
B
~
j
ε
,
ε
y
1
-
2
s
|
w
~
ε
|
2
γ
0
𝑑
x
𝑑
y
+
∫
Γ
~
j
ε
,
ε
|
w
~
ε
(
x
,
0
)
|
2
s
*
𝑑
x
≤
C
d
0
/
k
ε
→
0
as
ε
→
0
.
Choose cut-off functions {ξε,1} and {ξε,2} such that
ξ
ε
,
1
=
{
1
on
B
(
2
β
/
ε
)
+
(
5
j
ε
+
1
)
k
ε
+
(
0
)
¯
,
0
on
ℝ
+
N
+
1
∖
B
(
2
β
/
ε
)
+
(
5
j
ε
+
2
)
k
ε
+
(
0
)
,
ξ
ε
,
2
=
{
0
on
B
(
2
β
/
ε
)
+
(
5
j
ε
+
3
)
k
ε
+
(
0
)
¯
,
1
on
ℝ
+
N
+
1
∖
B
(
2
β
/
ε
)
+
(
5
j
ε
+
4
)
k
ε
+
(
0
)
,
0
≤
ξ
ε
,
1
,
ξ
ε
,
2
≤
1
, |∇ξε,1|,|∇ξε,2|≤C/kε. Furthermore, set w~ε,i:=ξε,iw~ε and wε,i(x,y):=w~ε,i(x-(xε/ε),y). Since wε∈Xε, we see that wε,i∈Xε, i=1,2. Thus (i)–(iii) hold. Direct computations show that
∥
w
ε
-
w
ε
,
1
-
w
ε
,
2
∥
X
ε
2
≤
C
∫
B
(
2
β
/
ε
)
+
(
5
j
ε
+
4
)
k
ε
+
(
0
)
∖
B
(
2
β
/
ε
)
+
(
5
j
ε
+
1
)
k
ε
+
(
0
)
y
1
-
2
s
|
∇
w
~
ε
|
2
𝑑
x
𝑑
y
+
C
∫
Γ
(
2
β
/
ε
)
+
(
5
j
ε
+
4
)
k
ε
0
(
0
)
∖
Γ
(
2
β
/
ε
)
+
(
5
j
ε
+
1
)
k
ε
0
(
0
)
V
(
ε
x
+
x
ε
)
|
w
~
ε
(
x
,
0
)
|
2
𝑑
x
+
C
∫
B
(
2
β
/
ε
)
+
(
5
j
ε
+
2
)
k
ε
+
(
0
)
∖
B
(
2
β
/
ε
)
+
(
5
j
ε
+
1
)
k
ε
+
(
0
)
y
1
-
2
s
|
∇
ξ
ε
,
1
|
2
|
w
~
ε
|
2
𝑑
x
𝑑
y
+
C
∫
B
(
2
β
/
ε
)
+
(
5
j
ε
+
4
)
k
ε
+
(
0
)
∖
B
(
2
β
/
ε
)
+
(
5
j
ε
+
3
)
k
ε
+
(
0
)
y
1
-
2
s
|
∇
ξ
ε
,
2
|
2
|
w
~
ε
|
2
𝑑
x
𝑑
y
=
(I)
+
(II)
+
(III)
+
(IV)
.
By (3.4), we obtain (I),(II)=o(1). Similar to the argument in (3.1), we see from (3.4) that
(III)
≤
C
(
∫
B
(
2
β
/
ε
)
+
(
5
j
ε
+
2
)
k
ε
+
(
0
)
∖
B
(
2
β
/
ε
)
+
(
5
j
ε
+
1
)
k
ε
+
(
0
)
y
1
-
2
s
|
w
~
ε
|
2
γ
0
𝑑
x
𝑑
y
)
1
/
γ
0
=
o
(
1
)
.
Similarly, (IV)=o(1). Hence, (iv) holds. Moreover, by (3.4), conditions (v) and (vi) hold. Thus we see from (i)–(vi), (f1), (f2) and the boundedness of {wε} that
∥
w
ε
∥
X
s
(
ℝ
+
N
+
1
)
2
=
∥
w
ε
,
1
∥
X
s
(
ℝ
+
N
+
1
)
2
+
∥
w
ε
,
2
∥
X
s
(
ℝ
+
N
+
1
)
2
+
o
(
1
)
,
∫
ℝ
N
V
(
ε
x
)
|
w
ε
(
x
,
0
)
|
2
𝑑
x
=
∫
ℝ
N
V
(
ε
x
)
|
w
ε
,
1
(
x
,
0
)
|
2
𝑑
x
+
∫
ℝ
N
V
(
ε
x
)
|
w
ε
,
2
(
x
,
0
)
|
2
𝑑
x
+
o
(
1
)
,
∫
ℝ
N
F
(
w
ε
(
x
,
0
)
)
𝑑
x
=
∫
ℝ
N
F
(
w
ε
,
1
(
x
,
0
)
)
𝑑
x
+
∫
ℝ
N
F
(
w
ε
,
2
(
x
,
0
)
)
𝑑
x
+
o
(
1
)
.
Hence, we get
(3.5)
J
ε
(
w
ε
)
≥
I
ε
(
w
ε
,
1
)
+
I
ε
(
w
ε
,
2
)
+
o
(
1
)
.
Next, we claim that ∥wε,2∥Xε→0 as ε→0. From (3.2), (iv) and the definition of wε,2 we see that
∥
w
ε
,
2
∥
X
ε
≤
∥
w
ε
,
1
-
ϕ
0
(
ε
(
|
x
-
(
x
ε
/
ε
)
|
2
+
y
2
)
1
/
2
/
β
)
W
0
(
x
-
(
x
ε
/
ε
)
,
y
)
∥
X
ε
+
2
d
0
+
o
(
1
)
=
∥
w
ε
,
1
-
ϕ
0
(
ε
(
|
x
-
(
x
ε
/
ε
)
|
2
+
y
2
)
1
/
2
/
β
)
W
0
(
x
-
(
x
ε
/
ε
)
,
y
)
∥
X
ε
(
B
(
2
β
/
ε
)
+
(
5
j
ε
+
2
)
k
ε
+
(
x
ε
/
ε
,
0
)
)
+
2
d
0
+
o
(
1
)
≤
∥
w
ε
,
2
∥
X
ε
(
B
(
2
β
/
ε
)
+
(
5
j
ε
+
2
)
k
ε
+
(
x
ε
/
ε
,
0
)
)
+
4
d
0
+
o
(
1
)
=
4
d
0
+
o
(
1
)
,
where o(1)→0 as ε→0. Hence,
lim
¯
ε
→
0
∥
w
ε
,
2
∥
X
ε
≤
4
d
0
.
Since 〈Jε′(wε),wε,2〉→0 as ε→0 and 〈Qε′(wε),wε,2〉≥0, we get from (f1) and (f2) that
k
s
∫
ℝ
+
N
+
1
y
1
-
2
s
∇
w
ε
⋅
∇
w
ε
,
2
d
x
d
y
+
∫
ℝ
N
V
(
ε
x
)
w
ε
(
x
,
0
)
w
ε
,
2
(
x
,
0
)
𝑑
x
≤
∫
ℝ
N
f
(
w
ε
(
x
,
0
)
)
w
ε
,
2
(
x
,
0
)
𝑑
x
+
o
(
1
)
≤
α
2
∫
ℝ
N
w
ε
(
x
,
0
)
w
ε
,
2
(
x
,
0
)
𝑑
x
+
C
∫
ℝ
N
|
w
ε
(
x
,
0
)
|
2
s
*
-
1
|
w
ε
,
2
(
x
,
0
)
|
𝑑
x
+
o
(
1
)
.
Then we get from (iii), (iv) and Lemma 2.2 that ∥wε,2∥Xε2≤C∥wε,2∥Xε2s*+o(1). Taking d0>0 small, we see that ∥wε,2∥Xε=o(1). Then by (3.5) we have
(3.6)
J
ε
(
w
ε
)
≥
I
ε
(
w
ε
,
1
)
+
o
(
1
)
.
Up to a subsequence, there exists w~∈X1,s(ℝ+N+1) such that
(3.7)
w
~
ε
,
1
⇀
w
~
in
X
1
,
s
(
ℝ
+
N
+
1
)
and
w
~
ε
,
1
(
x
,
0
)
→
w
~
(
x
,
0
)
a.e. in
ℝ
N
.
Next, we claim that
(3.8)
w
~
ε
,
1
(
x
,
0
)
→
w
~
(
x
,
0
)
in
L
2
s
*
(
ℝ
N
)
.
Indeed, by Lemma 2.4, we assume on the contrary that there exists r>0 such that
lim
¯
ε
→
0
sup
z
∈
ℝ
N
∫
Γ
1
0
(
z
)
|
w
~
ε
,
1
(
x
,
0
)
-
w
~
(
x
,
0
)
|
2
s
*
𝑑
x
=
2
r
>
0
.
Then, for ε>0 small, there exists zε∈ℝN such that
(3.9)
∫
Γ
1
0
(
z
ε
)
|
w
~
ε
,
1
(
x
,
0
)
-
w
~
(
x
,
0
)
|
2
s
*
𝑑
x
≥
r
>
0
.
There are two cases.
Case 1:
{
z
ε
}
is bounded, i.e. |zε|≤α0 for some α0>0. Then for ε>0 small,
(3.10)
∫
Γ
α
0
+
1
0
(
0
)
|
v
~
ε
(
x
,
0
)
|
2
s
*
𝑑
x
≥
r
>
0
,
where v~ε=w~ε,1-w~ and v~ε⇀0 in X1,s(ℝ+N+1). Next, we claim that
(3.11)
lim
ε
→
0
sup
φ
~
∈
X
0
s
(
B
α
0
+
2
+
(
0
)
)
,
∥
φ
~
∥
X
s
(
ℝ
+
N
+
1
)
=
1
|
〈
I
0
′
(
v
~
ε
)
,
φ
~
〉
|
=
0
,
where I0 is mentioned in (2.5). Since for ε>0 small, we have Γα0+20(xε/ε)⊂(Λ/ε) and ∥wε,2∥Xε→0 as ε→0, we see that
o
(
1
)
=
〈
J
ε
′
(
w
ε
)
,
φ
~
(
x
-
(
x
ε
/
ε
)
,
y
)
〉
(3.12)
=
k
s
∫
ℝ
+
N
+
1
∇
w
~
ε
,
1
⋅
∇
φ
~
+
∫
ℝ
N
V
(
ε
x
+
x
ε
)
w
~
ε
,
1
(
x
,
0
)
φ
~
(
x
,
0
)
𝑑
x
-
∫
ℝ
N
f
(
w
~
ε
,
1
(
x
,
0
)
)
φ
~
(
x
,
0
)
𝑑
x
+
o
(
1
)
,
where o(1)→0 as ε→0 uniformly for all φ~∈X0s(Bα0+2+(0)) with ∥φ~∥Xs(ℝ+N+1)=1.
Similar to (3.12) and the fact that xε→x0∈ℳβ as ε→0, we see that w~≥0 and it satisfies
(3.13)
{
-
div
(
y
1
-
2
s
∇
w
~
)
=
0
in
ℝ
+
N
+
1
,
-
k
s
lim
y
→
0
+
y
1
-
2
s
∂
w
~
∂
y
(
x
,
y
)
=
-
V
(
x
0
)
w
~
+
f
(
w
~
)
on
ℝ
N
×
{
0
}
.
Since f is of critical growth, similar to the arguments in [35, Lemma 4.7], we see that, as ε→0,
(3.14)
∫
ℝ
N
(
f
(
w
~
ε
,
1
(
x
,
0
)
)
-
f
(
v
~
ε
(
x
,
0
)
)
-
f
(
w
~
(
x
,
0
)
)
)
φ
~
(
x
,
0
)
𝑑
x
→
0
uniformly for all φ~∈X0s(Bα0+2+(0)) with ∥φ~∥Xs(ℝ+N+1)=1. From (3.12)–(3.14), we see that
〈
I
0
′
(
v
~
ε
)
,
φ
~
〉
=
-
(
∫
ℝ
N
V
(
ε
x
+
x
ε
)
w
~
ε
,
1
(
x
,
0
)
φ
~
(
x
,
0
)
𝑑
x
-
∫
ℝ
N
V
(
x
0
)
w
~
(
x
,
0
)
φ
~
(
x
,
0
)
𝑑
x
)
+
o
(
1
)
=
(
V
)
+
o
(
1
)
,
where o(1)→0 as ε→0 uniformly for all φ~∈X0s(Bα0+2+(0)) with ∥φ~∥Xs(ℝ+N+1)=1.
Since
sup
x
∈
Γ
α
0
+
2
0
(
0
)
V
(
ε
x
+
x
ε
)
≤
∥
V
∥
L
∞
(
Λ
)
for ε>0 small, xε→x0 as ε→0 and V(εx+xε)→V(x0) uniformly for x∈Γα0+20(0), we deduce that
|
(
V
)
|
≤
∫
ℝ
N
V
(
ε
x
+
x
ε
)
|
v
~
ε
(
x
,
0
)
φ
~
(
x
,
0
)
|
𝑑
x
+
∫
ℝ
N
|
V
(
ε
x
+
x
ε
)
-
V
(
x
0
)
|
⋅
|
w
~
(
x
,
0
)
φ
~
(
x
,
0
)
|
𝑑
x
≤
∥
V
∥
L
∞
(
Λ
)
∥
v
~
ε
(
x
,
0
)
∥
L
2
(
Γ
α
0
+
2
0
(
0
)
)
∥
φ
~
(
x
,
0
)
∥
L
2
(
Γ
α
0
+
2
0
(
0
)
)
+
o
(
1
)
∥
w
~
(
x
,
0
)
∥
L
2
(
Γ
α
0
+
2
0
(
0
)
)
∥
φ
~
(
x
,
0
)
∥
L
2
(
Γ
α
0
+
2
0
(
0
)
)
.
Lemma 2.3, (3.7) and (2.2) show that (V)→0 as ε→0 uniformly for all
φ
~
∈
X
0
s
(
B
α
0
+
2
+
(
0
)
)
with
∥
φ
~
∥
X
s
(
ℝ
+
N
+
1
)
=
1
.
Then (3.11) holds. By Lemma 2.7, we see from (3.10) and (3.11) that there exist z~ε∈Γα0+10(0)¯ and σε>0 with σε→0 as ε→0 such that
w
^
ε
(
x
,
y
)
:=
σ
ε
(
N
-
2
s
)
/
2
v
~
ε
(
σ
ε
(
x
,
y
)
+
(
z
~
ε
,
0
)
)
⇀
w
^
(
x
,
y
)
in
X
s
(
ℝ
+
N
+
1
)
,
where w^≥0 is a nontrivial solution of (2.6) and satisfies
(3.15)
∫
ℝ
+
N
+
1
k
s
y
1
-
2
s
|
∇
w
^
|
2
𝑑
x
𝑑
y
=
∫
ℝ
N
|
w
^
(
x
,
0
)
|
2
s
*
𝑑
x
=
(
k
s
S
(
s
,
N
)
)
N
/
(
2
s
)
.
Similar to the argument in (3.1), we see from (3.2) that
∫
ℝ
+
N
+
1
y
1
-
2
s
|
∇
w
ε
|
2
𝑑
x
𝑑
y
≤
C
d
0
+
∫
ℝ
+
N
+
1
y
1
-
2
s
|
∇
(
ϕ
0
(
ε
(
|
x
-
(
x
ε
/
ε
)
|
2
+
y
2
)
1
/
2
/
β
)
W
0
(
x
-
(
x
ε
/
ε
)
,
y
)
)
|
2
𝑑
x
𝑑
y
≤
C
d
0
+
∫
ℝ
+
N
+
1
y
1
-
2
s
|
∇
W
0
|
2
𝑑
x
𝑑
y
+
o
(
1
)
.
On the other hand,
∫
ℝ
+
N
+
1
y
1
-
2
s
|
∇
w
^
|
2
𝑑
x
𝑑
y
≤
lim
¯
ε
→
0
∫
ℝ
+
N
+
1
y
1
-
2
s
|
∇
w
^
ε
|
2
𝑑
x
𝑑
y
=
lim
¯
ε
→
0
∫
ℝ
+
N
+
1
y
1
-
2
s
|
∇
v
~
ε
|
2
𝑑
x
𝑑
y
=
lim
¯
ε
→
0
∫
ℝ
+
N
+
1
y
1
-
2
s
|
∇
w
~
ε
,
1
|
2
𝑑
x
𝑑
y
-
∫
ℝ
+
N
+
1
y
1
-
2
s
|
∇
w
~
|
2
𝑑
x
𝑑
y
≤
lim
¯
ε
→
0
∫
ℝ
+
N
+
1
y
1
-
2
s
|
∇
w
~
ε
|
2
𝑑
x
𝑑
y
.
Hence, we get
(3.16)
∫
ℝ
+
N
+
1
y
1
-
2
s
|
∇
w
^
|
2
𝑑
x
𝑑
y
≤
∫
ℝ
+
N
+
1
y
1
-
2
s
|
∇
W
0
|
2
𝑑
x
𝑑
y
+
C
d
0
.
Formulas (3.15) and (3.16) show that
c
V
0
=
I
V
0
(
W
0
)
-
1
N
P
V
0
(
W
0
)
=
s
N
k
s
∫
ℝ
+
N
+
1
y
1
-
2
s
|
∇
W
0
|
2
𝑑
x
𝑑
y
≥
s
N
k
s
∫
ℝ
+
N
+
1
y
1
-
2
s
|
∇
w
^
|
2
𝑑
x
𝑑
y
-
C
d
0
(3.17)
=
s
N
(
k
s
S
(
s
,
N
)
)
N
/
(
2
s
)
-
C
d
0
.
Letting d0→0 in (3.17), we obtain a contradiction to Proposition 2.8.
Case 2:
{
z
ε
}
is unbounded, i.e. up to a subsequence,
(3.18)
lim
ε
→
0
|
z
ε
|
=
∞
.
Then, by (3.9),
lim
¯
ε
→
0
∫
Γ
1
0
(
z
ε
)
|
w
~
ε
,
1
(
x
,
0
)
|
2
s
*
𝑑
x
≥
r
>
0
.
Since ξε,1(x,0)=0 for |x|≥(2β/ε)+(5jε+2)kε, we see that |zε|<(2β/ε)+(5jε+3)kε for ε>0 small. We assume that
ε
z
ε
→
z
0
∈
Γ
3
β
0
(
0
)
¯
and
w
¯
ε
(
x
,
y
)
:=
w
~
ε
,
1
(
x
+
z
ε
,
y
)
⇀
w
¯
(
x
,
y
)
in
X
1
,
s
(
ℝ
+
N
+
1
)
.
Similar to (3.12), we see that w¯ satisfies
{
-
div
(
y
1
-
2
s
∇
w
¯
)
=
0
in
ℝ
+
N
+
1
,
-
k
s
lim
y
→
0
+
y
1
-
2
s
∂
w
¯
∂
y
(
x
,
y
)
=
-
V
(
x
0
+
z
0
)
w
¯
+
f
(
w
¯
)
on
ℝ
N
×
{
0
}
.
If w¯≠0, we have
c
V
(
x
0
+
z
0
)
≤
s
N
k
s
∫
ℝ
+
N
+
1
y
1
-
2
s
|
∇
w
¯
|
2
𝑑
x
𝑑
y
.
Then, for large R>0,
lim
¯
ε
→
0
s
N
k
s
∫
B
R
+
(
z
ε
+
(
x
ε
/
ε
)
,
0
)
y
1
-
2
s
|
∇
w
ε
|
2
𝑑
x
𝑑
y
=
lim
¯
ε
→
0
s
N
k
s
∫
B
R
+
(
z
ε
+
(
x
ε
/
ε
)
,
0
)
y
1
-
2
s
|
∇
w
ε
,
1
|
2
𝑑
x
𝑑
y
=
lim
¯
ε
→
0
s
N
k
s
∫
B
R
+
(
0
)
y
1
-
2
s
|
∇
w
¯
ε
|
2
𝑑
x
𝑑
y
≥
s
N
k
s
∫
B
R
+
(
0
)
y
1
-
2
s
|
∇
w
¯
|
2
𝑑
x
𝑑
y
≥
1
2
[
s
N
k
s
∫
ℝ
+
N
+
1
y
1
-
2
s
|
∇
w
¯
|
2
𝑑
x
𝑑
y
]
(3.19)
≥
1
2
c
V
(
x
0
+
z
0
)
>
0
.
On the other hand, similar to the arguments in (3.1), we derive from (3.2) and (3.18) that
s
N
k
s
∫
B
R
+
(
z
ε
+
(
x
ε
/
ε
)
,
0
)
y
1
-
2
s
|
∇
w
ε
|
2
𝑑
x
𝑑
y
≤
C
∫
B
R
+
(
z
ε
+
(
x
ε
/
ε
)
,
0
)
y
1
-
2
s
|
∇
(
ϕ
0
(
ε
(
|
x
-
(
x
ε
/
ε
)
|
2
+
y
2
)
1
/
2
/
β
)
W
0
(
x
-
(
x
ε
/
ε
)
,
y
)
)
|
2
𝑑
x
+
C
d
0
≤
C
∫
B
R
+
(
z
ε
,
0
)
y
1
-
2
s
|
∇
W
0
|
2
𝑑
x
𝑑
y
+
C
ε
2
∫
B
R
+
(
z
ε
,
0
)
y
1
-
2
s
|
W
0
|
2
𝑑
x
𝑑
y
+
C
d
0
≤
C
∫
B
R
+
(
z
ε
,
0
)
y
1
-
2
s
|
∇
W
0
|
2
𝑑
x
𝑑
y
+
C
ε
2
R
2
(
∫
B
R
+
(
z
ε
,
0
)
y
1
-
2
s
|
W
0
|
2
γ
0
𝑑
x
𝑑
y
)
1
/
(
2
γ
0
)
+
C
d
0
(3.20)
=
C
d
0
+
o
(
1
)
,
which leads to a contradiction for d0>0 small.
If w¯≡0, then w¯ε⇀0 in X1,s(ℝ+N+1). Meanwhile,
(3.21)
lim
¯
ε
→
0
∫
Γ
1
0
(
0
)
|
w
¯
ε
(
x
,
0
)
|
2
s
*
𝑑
x
≥
r
>
0
,
and similar to Case 1, we can verify that
(3.22)
lim
ε
→
0
sup
φ
¯
∈
X
0
s
(
B
2
+
(
0
)
)
,
∥
φ
¯
∥
X
s
(
ℝ
+
N
+
1
)
=
1
|
〈
I
0
′
(
w
¯
ε
)
,
φ
¯
〉
|
=
0
.
Using Lemma 2.7 again, we see from (3.21) and (3.22) that there exists x~ε∈Γ10(0)¯ and γε>0 such that γε→0 as ε→0 and
w
ε
*
(
x
,
y
)
:=
γ
ε
(
N
-
2
s
)
/
2
w
¯
ε
(
γ
ε
(
x
,
y
)
+
(
x
~
ε
,
0
)
)
⇀
w
*
(
x
,
y
)
in
X
s
(
ℝ
+
N
+
1
)
,
where w*≥0 is a nontrivial solution of (2.6) and satisfies (3.15). Thus, there exists R>0 such that
(3.23)
∫
Γ
R
0
(
0
)
|
w
*
(
x
,
0
)
|
2
s
*
𝑑
x
≥
1
2
∫
ℝ
N
|
w
*
(
x
,
0
)
|
2
s
*
𝑑
x
≥
1
2
(
k
s
S
(
s
,
N
)
)
N
/
(
2
s
)
>
0
.
On the other hand, we see from the fact ∥wε,2∥Xε→0 as ε→0 that
∫
Γ
R
0
(
0
)
|
w
*
(
x
,
0
)
|
2
s
*
𝑑
x
≤
lim
¯
ε
→
0
∫
Γ
R
0
(
0
)
|
w
ε
*
(
x
,
0
)
|
2
s
*
𝑑
x
=
lim
¯
ε
→
0
∫
Γ
γ
ε
R
0
(
x
~
ε
)
|
w
¯
ε
(
x
,
0
)
|
2
s
*
𝑑
x
=
lim
¯
ε
→
0
∫
Γ
γ
ε
R
0
(
x
~
ε
+
z
ε
+
(
x
ε
/
ε
)
)
|
w
ε
(
x
,
0
)
|
2
s
*
𝑑
x
(3.24)
≤
lim
¯
ε
→
0
∫
Γ
2
0
(
z
ε
+
(
x
ε
/
ε
)
)
|
w
ε
(
x
,
0
)
|
2
s
*
𝑑
x
.
Similar to the arguments in (3.19) and (3.20), we see that (3.23) and (3.24) contradict (3.2) for d0>0 small. Therefore, (3.8) holds and the Interpolation Inequality shows that, as ε→0,
(3.25)
w
~
ε
,
1
(
x
,
0
)
→
w
~
(
x
,
0
)
in
L
q
(
ℝ
N
)
,
q
∈
(
2
,
2
s
*
]
.
By the fact that xε→x0∈ℳβ as ε→0 and (3.6) and (3.25), we get
(3.26)
I
V
(
x
0
)
(
w
~
)
≤
c
V
0
.
From the fact that ∥wε,2∥Xε→0 as ε→0 and (3.2) and (3.25), we see that w~≠0 for d0>0 small. Then, by (3.13),
(3.27)
I
V
(
x
0
)
(
w
~
)
≥
c
V
(
x
0
)
.
Since x0∈ℳβ⊂Λ, inequalities (3.26) and (3.27) imply that V(x0)=V0 and x0∈ℳ. At this point, it is clear that there exists W∈SV0 and z0∈ℝN such that w~(x,y)=W(x-z0,y). In view of (3.7), (3.25), (3.13), (iv) and the facts 〈Jε′(wε),wε,1〉→0, ∥wε,2∥Xε→0 as ε→0, 〈Qε′(wε),wε,1〉≡0 and w~ε,1(x,y)=wε,1(x+(xε/ε),y), we get
k
s
∫
ℝ
+
N
+
1
y
1
-
2
s
|
∇
w
~
|
2
𝑑
x
𝑑
y
+
∫
ℝ
N
V
(
x
0
)
|
w
~
(
x
,
0
)
|
2
𝑑
x
≤
lim
¯
ε
→
0
[
k
s
∫
ℝ
+
N
+
1
y
1
-
2
s
|
∇
w
~
ε
,
1
|
2
𝑑
x
𝑑
y
+
∫
ℝ
N
V
(
ε
x
+
x
ε
)
|
w
~
ε
,
1
(
x
,
0
)
|
2
𝑑
x
]
=
lim
¯
ε
→
0
∫
ℝ
N
f
(
w
~
ε
,
1
(
x
,
0
)
)
w
~
ε
,
1
(
x
,
0
)
𝑑
x
=
∫
ℝ
N
f
(
w
~
(
x
,
0
)
)
w
~
(
x
,
0
)
𝑑
x
=
k
s
∫
ℝ
+
N
+
1
y
1
-
2
s
|
∇
w
~
|
2
𝑑
x
𝑑
y
+
∫
ℝ
N
V
(
x
0
)
|
w
~
(
x
,
0
)
|
2
𝑑
x
.
Thus, as ε→0,
(3.28)
∫
ℝ
+
N
+
1
y
1
-
2
s
|
∇
w
~
ε
,
1
|
2
𝑑
x
𝑑
y
→
∫
ℝ
+
N
+
1
y
1
-
2
s
|
∇
w
~
|
2
𝑑
x
𝑑
y
and
(3.29)
∫
ℝ
N
V
(
ε
x
+
x
ε
)
|
w
~
ε
,
1
(
x
,
0
)
|
2
𝑑
x
→
∫
ℝ
N
V
(
x
0
)
|
w
~
(
x
,
0
)
|
2
𝑑
x
.
Since
V
(
x
0
)
=
V
0
≤
V
(
ε
x
+
x
ε
)
on
Γ
(
2
β
/
ε
)
+
(
5
j
ε
+
2
)
k
ε
0
(
0
)
,
by (3.28) and (3.29) we see that w~ε,1→w~ in X1,s(ℝ+N+1) as ε→0, which implies that
lim
ε
→
0
∥
w
ε
-
ϕ
0
(
ε
(
|
x
-
(
(
x
ε
/
ε
)
+
z
0
)
|
2
+
y
2
)
1
/
2
/
β
)
W
(
x
-
(
(
x
ε
/
ε
)
+
z
0
)
,
y
)
∥
X
ε
=
0
.
This completes the proof. ∎
By Lemma 3.2, the proof of the next lemma can be done as in [7, 21], and the details are omitted here.
Lemma 3.3.
There exists ε¯>0 such that for each ε∈(0,ε¯] there exists a sequence {wn,ε}n=1∞⊂Jεc~ε+ε∩Eεd0 such that Jε′(wn,ε)→0 in (Xε)-1 as n→∞, where (Xε)-1 is the dual space of Xε.
Proof.
See [7, Propositions 5–7] or [21, Lemmas 4.4–4.6]. ∎
Proof of Theorem 1.2.
By Lemma 3.3, there exists ε¯>0 such that for each ε∈(0,ε¯] there exists a sequence {wn,ε}n=1∞⊂Jεc~ε+ε∩Eεd0 such that Jε′(wn,ε)→0 in (Xε)-1 as n→∞. Since {wn,ε}n=1∞ is bounded in Xε, up to a subsequence, as n→∞, we have
(3.30)
{
w
n
,
ε
⇀
w
ε
in
X
ε
,
w
n
,
ε
(
x
,
0
)
→
w
ε
(
x
,
0
)
a.e. in
ℝ
N
and
(3.31)
λ
n
,
ε
:=
(
∫
ℝ
N
χ
ε
(
x
)
w
n
,
ε
2
(
x
,
0
)
𝑑
x
-
1
)
+
→
λ
ε
.
It is easy to verify that wε satisfies
(3.32)
{
-
div
(
y
1
-
2
s
∇
w
ε
)
=
0
in
ℝ
+
N
+
1
,
-
k
s
lim
y
→
0
+
y
1
-
2
s
∂
w
ε
∂
y
(
x
,
y
)
=
-
V
(
ε
x
)
w
ε
-
4
λ
ε
χ
ε
w
ε
+
g
ε
(
x
,
w
ε
)
on
ℝ
N
×
{
0
}
.
For any fixed ε∈(0,ε¯], choosing R>0 such that (Λ/ε)⊂ΓR0(0) and defining ϕR(x,y):=ϕ¯((|x|2+y2)1/2/R), y≥0, where ϕ¯∈C∞(ℝ+) satisfying ϕ¯(t)=0 if 0≤t≤1, ϕ¯(t)=1 if t≥2, 0≤ϕ¯(t)≤1 and |ϕ¯′(t)|≤C. Since {ϕRwn,ε}n=1∞ is bounded in Xε for each ε∈(0,ε¯], it follows that 〈Jε′(wn,ε),ϕRwn,ε〉→0 as n→∞. Then by the definition of gε(x,t) we have
k
s
∫
ℝ
+
N
+
1
y
1
-
2
s
|
∇
w
n
,
ε
|
2
ϕ
R
𝑑
x
𝑑
y
+
∫
ℝ
N
V
(
ε
x
)
w
n
,
ε
2
(
x
,
0
)
ϕ
R
(
x
,
0
)
𝑑
x
(3.33)
≤
κ
0
∫
ℝ
N
V
(
ε
x
)
w
n
,
ε
2
(
x
,
0
)
ϕ
R
(
x
,
0
)
𝑑
x
-
k
s
∫
ℝ
+
N
+
1
y
1
-
2
s
w
n
,
ε
(
∇
w
n
,
ε
⋅
∇
ϕ
R
)
𝑑
x
𝑑
y
.
Similar to (3.1), Hölder’s inequality, (3.30) and Lemma 2.1 (i) show that, as R→+∞,
lim
¯
n
→
∞
|
-
k
s
∫
ℝ
+
N
+
1
y
1
-
2
s
w
n
,
ε
(
∇
w
n
,
ε
⋅
∇
ϕ
R
)
𝑑
x
𝑑
y
|
≤
C
R
lim
¯
n
→
∞
(
∫
ℝ
+
N
+
1
y
1
-
2
s
|
∇
w
n
,
ε
|
2
𝑑
x
𝑑
y
)
1
/
2
lim
¯
n
→
∞
(
∫
B
2
R
+
(
0
)
∖
B
R
+
(
0
)
y
1
-
2
s
w
n
,
ε
2
𝑑
x
𝑑
y
)
1
/
2
≤
C
R
(
∫
B
2
R
+
(
0
)
∖
B
R
+
(
0
)
y
1
-
2
s
w
ε
2
𝑑
x
𝑑
y
)
1
/
2
(3.34)
≤
C
(
∫
B
2
R
+
(
0
)
∖
B
R
+
(
0
)
y
1
-
2
s
|
w
ε
|
2
γ
0
𝑑
x
𝑑
y
)
1
/
(
2
γ
0
)
→
0
.
Formulas (3.33) and (3.34) yield
lim
¯
R
→
+
∞
lim
¯
n
→
∞
[
k
s
∫
ℝ
+
N
+
1
∖
B
2
R
+
(
0
)
y
1
-
2
s
|
∇
w
n
,
ε
|
2
𝑑
x
𝑑
y
+
(
1
-
κ
0
)
∫
ℝ
N
∖
Γ
2
R
0
(
0
)
V
(
ε
x
)
w
n
,
ε
2
(
x
,
0
)
𝑑
x
]
=
0
,
which implies that the sequence {wn,ε}n=1∞ is tight in Xs(ℝ+N+1).
Next, we claim that, for ε>0 small,
(3.35)
w
n
,
ε
(
x
,
0
)
→
w
ε
(
x
,
0
)
in
L
2
s
*
(
ℝ
N
)
as
n
→
∞
.
Indeed, by Proposition 2.6, we may assume that
lim
n
→
∞
y
1
-
2
s
|
∇
w
n
,
ε
|
2
=
μ
and
lim
n
→
∞
|
w
n
,
ε
(
x
,
0
)
|
2
s
*
=
ν
in the sense of measures, and obtain an at most countable index set J, three families {xj}j∈J⊂ℝN, {μj}j∈J and {νj}j∈J with μj,νj≥0 such that
(3.36)
ν
=
|
w
ε
R
(
x
,
0
)
|
2
s
*
+
∑
j
∈
J
ν
j
δ
x
j
,
μ
≥
y
1
-
2
s
|
∇
w
ε
R
|
2
+
∑
j
∈
J
μ
j
δ
(
x
j
,
0
)
,
μ
j
≥
S
(
s
,
N
)
ν
j
2
/
2
s
*
,
where μ and ν are two bounded nonnegative measures on ℝ+N+1 and ℝN, respectively. It suffices to show that J=∅. Just suppose J≠∅, i.e. there exists xj0∈ℝN for some j0∈J. Similar to the arguments in Lemma 2.7, we get νj0≥(ksS(s,N))N/(2s). On the other hand, since {wn,ε}n=1∞⊂Eεd0, by the definition of Eεd0 there exist {Wn}n=1∞⊂SV0 and {xn}n=1∞⊂ℳβ such that
∥
w
n
,
ε
-
ϕ
0
(
ε
(
|
x
-
(
x
n
/
ε
)
|
2
+
y
2
)
1
/
2
/
β
)
W
n
(
x
-
(
x
n
/
ε
)
,
y
)
∥
X
ε
≤
3
2
d
0
.
By the compactness of SV0 and ℳβ there exist W0∈SV0 and x′∈ℳβ such that Wn→W0 in X1,s(ℝ+N+1) and xn→x′ as n→∞. Thus, for n large,
(3.37)
∥
w
n
,
ε
-
ϕ
0
(
ε
(
|
x
-
(
x
′
/
ε
)
|
2
+
y
2
)
1
/
2
/
β
)
W
0
(
x
-
(
x
′
/
ε
)
,
y
)
∥
X
ε
≤
2
d
0
.
We see from (3.36) and (3.37) that
c
~
ε
+
ε
≥
J
ε
(
w
n
,
ε
)
-
1
N
P
V
0
(
W
0
)
≥
s
N
k
s
∫
ℝ
+
N
+
1
y
1
-
2
s
|
∇
w
n
,
ε
|
2
𝑑
x
𝑑
y
+
k
s
2
(
∫
ℝ
+
N
+
1
y
1
-
2
s
|
∇
w
n
,
ε
|
2
𝑑
x
𝑑
y
-
∫
ℝ
+
N
+
1
y
1
-
2
s
|
∇
W
0
|
2
𝑑
x
𝑑
y
)
+
1
2
(
∫
ℝ
N
V
(
ε
x
)
(
w
n
,
ε
(
x
,
0
)
)
2
𝑑
x
-
∫
ℝ
N
V
0
(
W
0
(
x
,
0
)
)
2
𝑑
x
)
-
(
∫
ℝ
N
F
(
w
n
,
ε
(
x
,
0
)
)
𝑑
x
-
∫
ℝ
N
F
(
W
0
(
x
,
0
)
)
𝑑
x
)
≥
s
N
k
s
μ
j
0
-
C
d
0
+
o
(
1
)
≥
s
N
k
s
S
(
s
,
N
)
ν
j
0
2
/
2
s
*
-
C
d
0
+
o
(
1
)
≥
s
N
(
k
s
S
(
s
,
N
)
)
N
/
(
2
s
)
-
C
d
0
+
o
(
1
)
,
where o(1)→0 as ε→0. Taking ε→0 and d0→0, by Proposition 3.1 we have
c
V
0
≥
s
N
(
k
s
S
(
s
,
N
)
)
N
/
(
2
s
)
,
which contradicts Proposition 2.8. Therefore, (3.35) holds and the Interpolation Inequality shows that wn,ε(x,0)→wε(x,0) in Lq(ℝN), q∈(2,2s*], as n→∞. It follows from the definition of gε(x,t), (f1) and (f2) that, for ε>0 small,
(3.38)
∫
ℝ
N
g
ε
(
x
,
w
n
,
ε
(
x
,
0
)
)
w
n
,
ε
(
x
,
0
)
𝑑
x
→
∫
ℝ
N
g
ε
(
x
,
w
ε
(
x
,
0
)
)
w
ε
(
x
,
0
)
𝑑
x
,
n
→
∞
.
In view of (3.30), (3.31), (3.38), (3.32) and the fact 〈Jε′(wn,ε),wn,ε〉→0, we get that, for ε>0 small,
w
n
,
ε
→
w
ε
in
X
ε
,
n
→
∞
,
and
λ
ε
=
(
∫
ℝ
N
χ
ε
(
x
)
w
ε
2
(
x
,
0
)
𝑑
x
-
1
)
+
.
Since SV0 is compact in X1,s(ℝ+N+1), it is easy to see that 0∉Eεd0 for ε>0, d0>0 small. Thus, wε∈Eεd0∩Jεc~ε+ε is a nontrivial solution of (3.32).
For any sequence {εj}j=1∞ with εj→0 as j→∞ by Lemma 3.2 there exist, up to a subsequence, {zj}j=1∞⊂ℝN, x0∈ℳ and W∈SV0 such that
(3.39)
lim
j
→
∞
|
ε
j
z
j
-
x
0
|
=
0
and
lim
j
→
∞
∥
w
ε
j
(
x
,
y
)
-
ϕ
0
(
ε
j
(
|
x
-
z
j
|
2
+
y
2
)
1
/
2
/
β
)
W
(
x
-
z
j
,
y
)
∥
X
ε
j
=
0
,
which implies that w¯εj(x,0)→W(x,0) in L2s*(ℝN), where w¯εj(x,y):=wεj(x+zj,y). Similar to (2.21), we conclude that
(3.40)
lim
|
x
|
→
+
∞
w
¯
ε
j
(
x
,
0
)
=
0
uniformly for
ε
j
small
,
which implies that there is a ρ>0 such that f(w¯εj(x,0))≤κ0αw¯εj(x,0) for all |z|≥ρ and εj small. On the other hand, if |z|≤ρ, it follows from (3.39) that Γεjρ0(εjzj)⊂Λ for εj small. Therefore,
(3.41)
g
ε
j
(
x
+
z
j
,
w
¯
ε
j
(
x
,
0
)
)
=
f
(
w
¯
ε
j
(
x
,
0
)
)
for
ε
j
small
.
From (3.40) and (f1) we obtain for some large R>0,
f
(
w
¯
ε
j
(
x
,
0
)
)
≤
1
2
V
(
ε
j
x
+
ε
j
z
j
)
w
¯
ε
j
(
x
,
0
)
for
x
∈
ℝ
N
∖
Γ
R
0
(
0
)
.
The comparison arguments show that (see also [20])
w
¯
ε
j
(
x
,
0
)
≤
C
1
+
|
x
|
N
+
2
s
for
ε
j
small
,
where C is independent of εj. Thus, as j→∞,
ε
j
-
1
∫
ℝ
N
∖
(
Λ
/
ε
j
)
w
ε
j
2
(
x
,
0
)
𝑑
x
=
ε
j
-
1
∫
ℝ
N
∖
(
Λ
/
ε
j
-
z
j
)
w
¯
ε
j
2
(
x
,
0
)
𝑑
x
≤
C
ε
j
-
1
∫
ℝ
N
∖
Γ
β
/
ε
j
0
(
0
)
1
(
1
+
|
x
|
N
+
2
s
)
2
𝑑
x
→
0
,
that is, Qεj(wεj)=0 for εj small. Then, by (3.41), wεj is a solution of (1.4). Thus uεj(x):=wεj(x/εj,0) is a solution of (1.1). Since uε∈L∞(ℝN), uε is nonnegative and nontrivial, and V and f are continuous, we use Harnack’s inequality in [9, Lemma 4.9] to conclude that uε is positive.
Let Pj be a maximum point of w¯εj(x,0). Similar to (2.27), we verify that there exists C0>0 such that w¯εj(Pj,0)>C0 by (3.40), and {Pj}j=1∞ must be bounded. Since uεj(x)=w¯εj((x/εj)-zj,0), we obtain that xj:=εjPj+εjzj is a maximum point of uεj. By (3.39), xj→x0∈ℳ as j→∞. Since the sequence {εj}j=1∞ is arbitrary, we have obtained the existence and concentration results in Theorem 1.2.
To complete the proof, we only need to prove the polynomial decay of uε. Since the proof is standard (see [20] for example), we omit it here. ∎
