1 Introduction and main results
In this paper, we consider the following nonlinear Schrödinger equations involving the fractional p-Laplacian
(1.1)
(
−
Δ
)
p
s
u
(
x
)
+
λ
V
(
x
)
u
(
x
)
p
−
1
=
u
(
x
)
q
−
1
,
x
∈
R
N
,
u
(
x
)
≥
0
,
u
(
x
)
∈
W
s
,
p
(
R
N
)
,
where
λ
>
0
is a parameter,
1
<
p
<
q
<
N
p
N
−
s
p
,
N
≥
2
, and
V
(
x
)
is a real continuous function on
R
N
.
We are interested in the existence of ground state solutions for
λ
big enough, and their asymptotical behavior as
λ
→
∞
.
As far as we know, these kinds of problems were first put forward in [1] by Bartsch and Wang, where they studied the Schrödinger equations. Under suitable conditions imposed on the potential, the loss of compactness caused by the whole space
R
N
can be recovered when parameter
λ
is big enough. Then, many authors began studying the problems with potential well. A lot of results have been obtained.
Bartsch and Parnet [2] also considered the nonlinear Schrödinger equation:
−
Δ
u
+
(
a
0
(
x
)
+
λ
a
(
x
)
)
u
=
f
(
x
,
u
)
,
x
∈
R
N
,
u
(
x
)
→
0
,
∣
x
∣
→
∞
,
where
a
0
(
x
)
+
λ
a
(
x
)
is indefinite. By using a local linking theorem and the critical groups theory, they obtained the existence of solutions and their asymptotical behavior as
λ
→
∞
.
Xu and Chen [3] studied the following Kirchhoff problem:
−
a
+
b
∫
R
N
∣
∇
u
∣
2
d
x
Δ
u
+
λ
V
(
x
)
u
=
f
(
x
,
u
)
,
x
∈
R
N
,
u
(
x
)
∈
H
1
(
R
N
)
,
where
f
(
x
,
u
)
can be sublinear or superlinear. By using the genus theory, they obtained infinitely many negative solutions.
Aleves et al. [4] dealt with the following Choquard equation:
−
Δ
u
+
(
λ
a
(
x
)
+
1
)
u
=
1
∣
x
∣
μ
∗
∣
u
∣
p
∣
u
∣
p
−
2
u
,
x
∈
R
N
,
u
(
x
)
∈
H
1
(
R
N
)
,
where
μ
∈
(
0
,
3
)
,
p
∈
(
2
,
6
−
μ
)
, and the potential well
Ω
=
⋃
j
=
1
k
Ω
j
. They proved the existence of a solution, which is nonzero on any subset
Ω
j
. Furthermore, its asymptotical behavior was investigated.
Zhao et al. [5] studied the Schrödinger-Poisson system allowing the potential
V
(
x
)
changes sign
−
Δ
u
+
λ
V
(
x
)
u
+
K
(
x
)
ϕ
u
=
∣
u
∣
p
−
2
u
in
R
3
,
−
Δ
ϕ
=
K
(
x
)
u
2
in
R
3
,
where
p
∈
(
3
,
6
)
and
V
∈
C
(
R
3
,
R
)
are bounded from below. Using the variational method, they obtained the existence and asymptotic behavior of nontrivial solutions.
For the critical problems, Clapp and Ding [6] have studied the nonlinear Schrödinger equation:
−
Δ
u
+
λ
V
(
x
)
u
=
μ
u
+
u
2
∗
−
1
,
x
∈
R
N
for
N
≥
4
,
λ
,
μ
>
0
. By using variational methods, the authors established existence and multiplicity of positive solutions, which localize near the potential well for
λ
large and
μ
small.
Later, the corresponding results obtained in [6] were generalized to the fractional Schrödinger equations by Niu and Tang [7], where they have studied
(
−
Δ
)
s
u
+
(
λ
V
(
x
)
−
μ
)
u
=
∣
u
∣
2
s
∗
−
2
u
,
x
∈
R
N
,
u
≥
0
,
u
∈
H
s
(
R
N
)
.
Under the linear perturbation, [6] and [7] obtained the existence of solutions and their asymptotic behavior. For the nonlinear perturbation, Alves and Barros [8] considered
−
Δ
u
+
λ
V
(
x
)
u
=
μ
u
p
−
1
+
u
2
∗
−
1
,
x
∈
R
N
.
By employing the Ljusternik-Schnirelmann category, for
λ
big enough and
μ
small enough, the aforementioned problem has at last
cat
(
Ω
)
positive solutions.
For more results about these kinds of problems and fractional Schrödinger equations, see, for example, [9,10, 11,12,13, 14,15,16, 17,18,19, 20,21,22, 23,24] and references therein. Motivated by the aforementioned results, we consider equation (1.1). The potential function
V
(
x
)
satisfies
V
(
x
)
∈
C
(
R
N
,
R
)
such that
V
(
x
)
≥
0
,
Ω
≔
int
V
−
1
(
0
)
is a nonempty open set of class
C
0.1
with bounded boundary and
V
−
1
(
0
)
=
Ω
¯
;
There exists
M
0
>
0
such that
μ
(
{
x
∈
R
N
:
V
(
x
)
≤
M
0
}
)
<
∞
,
where
μ
denotes the Lebesgue measure on
R
N
.
We first introdcue some notations. For
s
∈
(
0
,
1
)
,
p
∈
[
1
,
+
∞
)
, define
W
s
,
p
(
R
N
)
≔
u
∈
L
p
(
R
N
)
:
∣
u
(
x
)
−
u
(
y
)
∣
∣
x
−
y
∣
N
p
+
s
∈
L
p
(
R
N
×
R
N
)
,
endowed with the norm
‖
u
‖
W
s
,
p
(
R
N
)
≔
∫
R
N
∣
u
∣
p
d
x
+
∫
R
N
∫
R
N
∣
u
(
x
)
−
u
(
y
)
∣
p
∣
x
−
y
∣
N
+
s
p
d
x
d
y
1
p
,
where the term
[
u
]
W
s
,
p
(
R
N
)
≔
∫
R
N
∫
R
N
∣
u
(
x
)
−
u
(
y
)
∣
p
∣
x
−
y
∣
N
+
p
s
d
x
d
y
1
p
is the so-called Gagliardo (semi)norm of
u
. Moreover, we define
W
s
,
p
(
Ω
)
≔
u
∈
L
p
(
Ω
)
:
∣
u
(
x
)
−
u
(
y
)
∣
∣
x
−
y
∣
N
p
+
s
∈
L
p
(
Ω
×
Ω
)
,
endowed with the norm
‖
u
‖
W
s
,
p
(
Ω
)
≔
∫
Ω
∣
u
∣
p
d
x
+
∫
Ω
∫
Ω
∣
u
(
x
)
−
u
(
y
)
∣
p
∣
x
−
y
∣
N
+
s
p
d
x
d
y
1
p
.
Let
E
λ
≔
u
∈
W
s
,
p
(
R
N
)
:
∫
R
N
λ
V
(
x
)
∣
u
(
x
)
∣
p
d
x
<
∞
,
with the norm
‖
u
‖
λ
=
∫
R
N
∫
R
N
∣
u
(
x
)
−
u
(
y
)
∣
p
∣
x
−
y
∣
N
+
p
s
d
x
d
y
+
∫
R
N
λ
V
(
x
)
∣
u
(
x
)
∣
p
d
x
1
p
.
The energy functional associated with (1.1) is
(1.2)
J
λ
(
u
)
=
1
p
∫
R
N
∫
R
N
∣
u
(
x
)
−
u
(
y
)
∣
p
∣
x
−
y
∣
N
+
p
s
d
x
d
y
+
λ
p
∫
R
N
V
(
x
)
∣
u
(
x
)
∣
p
d
x
−
1
q
∫
R
N
u
+
(
x
)
q
d
x
for
u
∈
E
λ
,
where
u
+
=
max
{
u
,
0
}
. Then, we can define the Nehari manifold
ℳ
λ
≔
u
∈
E
λ
⧹
{
0
}
:
∫
R
N
∫
R
N
∣
u
(
x
)
−
u
(
y
)
∣
p
∣
x
−
y
∣
N
+
p
s
d
x
d
y
+
λ
∫
R
N
V
(
x
)
∣
u
(
x
)
∣
p
d
x
=
∫
R
N
u
+
(
x
)
q
d
x
and
c
λ
≔
inf
{
J
λ
(
u
)
:
u
∈
ℳ
λ
}
.
Consider the following “limit” problem of (1.1)
(1.3)
(
−
Δ
)
p
s
u
(
x
)
=
u
(
x
)
q
−
1
,
x
∈
Ω
,
u
(
x
)
≥
0
,
x
∈
Ω
,
u
(
x
)
=
0
,
x
∈
R
N
⧹
Ω
,
Define a subspace
E
0
of
W
s
,
p
(
R
N
)
as follows:
(1.4)
E
0
≔
{
u
∈
W
s
,
p
(
R
N
)
:
u
(
x
)
=
0
in
R
N
⧹
Ω
}
tr
Ω
E
0
=
{
u
∣
Ω
:
u
∈
E
0
}
.
The energy functional associated with (1.3) can be defined by
Φ
(
u
)
=
1
p
∫
R
N
∫
R
N
∣
u
(
x
)
−
u
(
y
)
∣
p
∣
x
−
y
∣
N
+
p
s
d
x
d
y
−
1
q
∫
Ω
u
+
(
x
)
q
d
x
for
u
∈
E
0
.
Then, the associated Nehari manifold is
N
≔
u
∈
E
0
⧹
{
0
}
:
∫
R
N
∫
R
N
∣
u
(
x
)
−
u
(
y
)
∣
p
∣
x
−
y
∣
N
+
p
s
d
x
d
y
=
∫
Ω
u
+
(
x
)
q
d
x
and
c
(
Ω
)
≔
inf
{
Φ
(
u
)
:
u
∈
N
}
.
Definition 1.1
A function
u
λ
(
x
)
is a ground state solution of (1.1) if
c
λ
is achieved by
u
λ
∈
ℳ
λ
, which is a critical point of
J
λ
. Similarly, a function
u
(
x
)
is a ground state solution of (1.3) if
c
(
Ω
)
is achieved by
u
∈
N
, which is a critical point of
Φ
.
Definition 1.2
Let
X
be a Banach space,
φ
∈
C
1
(
X
,
R
)
.
The function
φ
satisfies the
(
P
S
)
c
condition if any sequence
{
u
n
}
⊆
X
, such that
(1.5)
φ
(
u
n
)
→
c
,
φ
′
(
u
n
)
→
0
has a convergent subsequence. The sequence
{
u
n
}
that satisfies (1.5) is called to be a
(
P
S
)
c
sequence of
φ
.
Our main results read as follows:
Theorem 1.3
Suppose
(
V
1
)
and
(
V
2
)
hold, then for
λ
large, (1.1) has a ground state solution
u
λ
(
x
)
. Furthermore, any sequence
λ
n
→
∞
,
{
u
λ
n
(
x
)
}
has a subsequence such that
u
λ
n
converges in
W
s
,
p
(
R
N
)
along the subsequence to a ground state solution
u
of (1.3).
Theorem 1.4
Suppose
(
V
1
)
and
(
V
2
)
hold. Let
u
n
,
n
∈
N
be a sequence of solutions of (1.1) with
λ
being replaced by
λ
n
(
λ
n
→
∞
as
n
→
∞
) such that
lim sup
n
→
∞
J
λ
(
u
n
)
<
∞
. Then,
u
n
(
x
)
converges strongly along a subsequence in
W
s
,
p
(
R
N
)
to a solution
u
of (1.3).
The following paper is organized as follows: In Section 2, we will give some preliminary results. Section 3 is devoted to the “limit” problem, and Section 4 contains the proofs of the main results.
C
denotes various generic positive constants, and
o
(
1
)
will be used to represent quantities that tend to 0 as
λ
(
or
n
)
→
∞
.
2 Preliminary results
Lemma 2.1
Let
λ
0
>
0
be a fixed constant. Then, for
λ
≥
λ
0
>
0
,
V
(
x
)
satisfying
(
V
1
)
and
(
V
2
)
,
E
λ
is continuously embedded in
W
s
,
p
(
R
N
)
uniformly in
λ
.
Proof
By the definition of
W
s
,
p
(
R
N
)
and
E
λ
, we only need to prove the following inequality:
(2.1)
∫
R
N
∣
u
(
x
)
∣
p
d
x
≤
C
∫
R
N
∫
R
N
∣
u
(
x
)
−
u
(
y
)
∣
p
∣
x
−
y
∣
N
+
p
s
d
x
d
y
+
∫
R
N
λ
V
(
x
)
∣
u
(
x
)
∣
p
d
x
.
Define
D
≔
{
x
∈
R
N
:
V
(
x
)
≤
M
0
}
and
D
δ
0
≔
{
x
∈
R
N
:
dist
(
x
,
D
)
≤
δ
0
}
.
Take
ζ
∈
C
∞
(
R
N
,
R
)
,
0
≤
ζ
≤
1
, satisfying
(2.2)
ζ
(
x
)
=
1
,
x
∈
D
,
0
,
x
∉
D
δ
0
,
∣
∇
ζ
∣
≤
C
/
δ
0
.
Then, for any function
u
∈
E
λ
, we can obtain
(2.3)
∫
R
N
(
1
−
ζ
p
)
∣
u
(
x
)
∣
p
d
x
=
∫
R
N
⧹
D
(
1
−
ζ
p
)
∣
u
(
x
)
∣
p
d
x
+
∫
D
(
1
−
ζ
p
)
∣
u
(
x
)
∣
p
d
x
≤
1
λ
0
M
0
λ
∫
R
N
V
(
x
)
∣
u
(
x
)
∣
p
d
x
and
(2.4)
∫
R
N
ζ
p
∣
u
(
x
)
∣
p
d
x
=
∫
D
δ
0
ζ
p
∣
u
(
x
)
∣
p
d
x
≤
μ
(
D
δ
0
)
1
−
p
p
s
∗
∫
D
δ
0
∣
u
(
x
)
∣
p
s
∗
d
x
p
p
s
∗
≤
C
μ
(
D
δ
0
)
1
−
p
p
s
∗
∫
R
N
∫
R
N
∣
u
(
x
)
−
u
(
y
)
∣
p
∣
x
−
y
∣
N
+
p
s
d
x
d
y
,
where we have used
(
V
2
)
and the Sobolev trace inequality
∫
R
N
∣
u
(
x
)
∣
p
s
∗
d
x
1
/
p
s
∗
≤
C
∫
R
N
∫
R
N
∣
u
(
x
)
−
u
(
y
)
∣
p
∣
x
−
y
∣
N
+
p
s
d
x
d
y
1
/
p
,
for
u
∈
W
s
,
p
(
R
N
)
and
C
=
C
(
N
,
p
,
s
)
>
0
. Thus, (2.1) follows from (2.3) and (2.4).□
Lemma 2.2
There exists
σ
>
0
independent of
λ
, such that
‖
u
‖
λ
≥
σ
for all
u
∈
ℳ
λ
.
Proof
From Lemma 2.1, for any
u
∈
ℳ
λ
,
0
=
⟨
J
λ
′
(
u
)
,
u
⟩
=
∫
R
N
∫
R
N
∣
u
(
x
)
−
u
(
y
)
∣
p
∣
x
−
y
∣
N
+
p
s
d
x
d
y
+
λ
∫
R
N
V
(
x
)
∣
u
(
x
)
∣
p
d
x
−
∫
R
N
u
+
(
x
)
q
d
x
≥
‖
u
‖
λ
p
−
C
‖
u
‖
W
s
,
p
(
R
N
)
q
≥
‖
u
‖
λ
p
−
C
‖
u
‖
λ
q
,
where
C
>
0
is independent of
λ
≥
0
. The aforementioned inequality implies that
‖
u
‖
λ
q
−
p
≥
1
C
. Choosing
σ
=
1
C
1
q
−
p
, we obtain
‖
u
‖
λ
≥
σ
.□
Lemma 2.3
Let
λ
0
be a fixed positive constant, there exists
c
0
>
0
independent of
λ
≥
λ
0
>
0
, such that if
{
u
n
}
is a
(
P
S
)
c
sequence of
J
λ
, then either
c
≥
c
0
or
c
=
0
. Moreover,
(2.5)
lim sup
n
→
∞
‖
u
n
‖
λ
p
≤
p
q
q
−
p
c
.
Proof
From the definition of
(
P
S
)
c
sequence,
c
+
‖
u
n
‖
λ
⋅
o
(
1
)
=
J
λ
(
u
n
)
−
1
q
⟨
J
λ
′
(
u
n
)
,
u
n
⟩
=
1
p
−
1
q
∫
R
N
∫
R
N
∣
u
n
(
x
)
−
u
n
(
y
)
∣
p
∣
x
−
y
∣
N
+
p
s
d
x
d
y
+
λ
∫
R
N
V
(
x
)
∣
u
n
(
x
)
∣
p
d
x
=
q
−
p
p
q
‖
u
n
‖
λ
p
.
Then, (2.5) holds. On the other side, there is a constant
C
>
0
independent of
λ
≥
λ
0
>
0
, such that
⟨
J
λ
′
(
u
)
,
u
⟩
=
∫
R
N
∫
R
N
∣
u
(
x
)
−
u
(
y
)
∣
p
∣
x
−
y
∣
N
+
p
s
d
x
d
y
+
λ
∫
R
N
V
(
x
)
∣
u
(
x
)
∣
p
d
x
−
∫
R
N
u
+
(
x
)
q
d
x
≥
‖
u
‖
λ
p
−
C
‖
u
‖
λ
q
.
Thus, there exists
σ
1
>
0
independent of
λ
, such that
(2.6)
1
4
‖
u
‖
λ
p
≤
⟨
J
λ
′
(
u
)
,
u
⟩
for
‖
u
‖
λ
<
σ
1
.
If
c
<
σ
1
p
(
q
−
p
)
p
q
, then
lim sup
n
→
∞
‖
u
n
‖
λ
p
≤
c
p
q
q
−
p
<
σ
1
p
.
Hence,
‖
u
n
‖
λ
<
σ
1
for
n
large. It follows from (2.6) that
1
4
‖
u
n
‖
λ
p
≤
⟨
J
λ
′
(
u
n
)
,
u
n
⟩
=
o
(
1
)
‖
u
n
‖
λ
,
which implies
‖
u
n
‖
λ
→
0
as
n
→
∞
. Therefore,
J
λ
(
u
n
)
→
0
, that is,
c
=
0
. Thus,
c
0
=
σ
1
p
(
q
−
p
)
q
p
is as required.□
Lemma 2.4
There exists
δ
0
>
0
, such that any
(
P
S
)
c
sequence
{
u
n
}
of
J
λ
with
λ
≥
0
and
c
>
0
satisfies
(2.7)
lim inf
n
→
∞
‖
u
n
+
‖
L
q
(
R
N
)
q
≥
δ
0
c
.
Proof
From the definition of
(
P
S
)
c
sequence,
c
=
lim
n
→
∞
J
λ
(
u
n
)
−
1
p
⟨
J
λ
′
(
u
n
)
,
u
n
⟩
=
1
p
−
1
q
lim
n
→
∞
∫
R
N
u
n
+
(
x
)
q
d
x
=
(
q
−
p
)
q
p
lim
n
→
∞
‖
u
n
+
(
x
)
‖
L
q
(
R
N
)
q
,
which implies (2.7) with
δ
0
≤
q
p
q
−
p
.□
Lemma 2.5
Let
C
1
be any fixed constant. Then, for any
ε
>
0
, there exists
Λ
ε
>
0
and
R
ε
>
0
, such that if
{
u
n
}
is a
(
P
S
)
c
sequence of
J
λ
with
λ
≥
Λ
ε
,
c
≤
C
1
, then
(2.8)
lim sup
n
→
∞
∫
B
R
ε
c
u
n
+
(
x
)
q
d
x
≤
ε
,
where
B
R
ε
c
=
{
x
∈
R
N
:
∣
x
∣
≥
R
ε
}
.
Proof
For
R
>
0
, let
A
(
R
)
≔
{
x
∈
R
N
:
∣
x
∣
>
R
,
V
(
x
)
≥
M
0
}
and
B
(
R
)
≔
{
x
∈
R
N
:
∣
x
∣
>
R
,
V
(
x
)
<
M
0
}
.
It follows from Lemma 2.3 that
(2.9)
∫
A
(
R
)
∣
u
n
(
x
)
∣
p
d
x
≤
1
λ
M
0
∫
R
N
λ
V
(
x
)
∣
u
n
(
x
)
∣
p
d
x
≤
1
λ
M
0
∫
R
N
∫
R
N
∣
u
n
(
x
)
−
u
n
(
y
)
∣
p
∣
x
−
y
∣
N
+
p
s
d
x
d
y
+
∫
R
N
λ
V
(
x
)
∣
u
n
(
x
)
∣
p
d
x
≤
1
λ
M
0
p
q
q
−
p
C
1
+
o
(
1
)
.
From Hölder inequality and (2.5), we can see that, for
1
<
r
<
N
/
(
N
−
p
s
)
,
(2.10)
∫
B
(
R
)
∣
u
n
(
x
)
∣
p
d
x
≤
∫
R
N
∣
u
n
(
x
)
∣
p
r
d
x
1
/
r
μ
(
B
(
R
)
)
1
/
r
′
≤
C
‖
u
n
‖
λ
p
⋅
μ
(
B
(
R
)
)
1
/
r
′
≤
C
p
q
q
−
p
C
0
⋅
μ
(
B
(
R
)
)
1
/
r
′
,
where
C
=
C
(
N
,
r
)
>
0
and
1
/
r
+
1
/
r
′
=
1
. By interpolation inequality and Sobolev embedding inequality, we can obtain
∫
B
R
c
u
n
+
(
x
)
q
d
x
≤
∫
B
R
c
∣
u
n
(
x
)
∣
p
d
x
q
(
1
−
θ
)
p
⋅
∫
B
R
c
∣
u
n
(
x
)
∣
p
s
∗
d
x
q
θ
p
s
∗
≤
∫
B
R
c
∣
u
n
(
x
)
∣
p
d
x
q
(
1
−
θ
)
p
∫
R
N
∣
u
n
(
x
)
∣
p
s
∗
d
x
q
θ
p
s
∗
≤
C
∫
B
R
c
∣
u
n
(
x
)
∣
p
d
x
q
(
1
−
θ
)
p
∫
R
N
∫
R
N
∣
u
n
(
x
)
−
u
n
(
y
)
∣
p
∣
x
−
y
∣
N
+
p
s
d
x
d
y
q
θ
p
≤
C
∫
A
(
R
)
∣
u
n
(
x
)
∣
p
d
x
+
∫
B
(
R
)
∣
u
n
(
x
)
∣
p
d
x
q
(
1
−
θ
)
p
‖
u
n
‖
λ
q
θ
,
where
θ
=
N
s
q
−
p
p
q
. Then, the result follows from (2.9), (2.10) and
(
V
2
)
.□
Lemma 2.6
(Brézis-Lieb lemma, 1983) Let
{
u
n
}
⊂
L
p
(
R
N
)
,
1
≤
p
<
∞
. If
{
u
n
}
is bounded in
L
p
(
R
N
)
,
u
n
→
u
almost everywhere on
R
N
, then
(2.11)
lim
n
→
∞
(
∣
u
n
∣
p
p
−
∣
u
n
−
u
∣
p
p
)
=
∣
u
∣
p
p
.
Lemma 2.7
Let
λ
≥
λ
0
>
0
be fixed and let
{
u
n
}
be a
(
P
S
)
c
sequence of
J
λ
. Then, up to a subsequence,
u
n
⇀
u
in
E
λ
with
u
being a weak solution of (1.1). Moreover,
u
n
1
=
u
n
−
u
is
(
P
S
)
c
′
sequence with
c
′
=
c
−
J
λ
(
u
)
.
Proof
By Lemma 2.3,
{
u
n
}
is bounded in
E
λ
. Then, up to a subsequence
u
n
⇀
u
in
E
λ
as
n
→
∞
, and
(2.12)
u
n
⇀
u
in
W
s
,
p
(
R
N
)
,
(2.13)
u
n
⇀
u
in
L
q
(
R
N
)
,
p
≤
q
<
p
s
∗
,
(2.14)
u
n
→
u
in
L
loc
q
(
R
N
)
,
p
≤
q
<
p
s
∗
,
(2.15)
u
n
→
u
a.e. in
R
N
,
where
p
s
∗
=
N
p
N
−
p
s
is the fractional critical Sobolev exponent. Hence, for any
φ
∈
E
λ
, we have
⟨
J
λ
′
(
u
n
)
,
φ
⟩
=
∫
R
N
∫
R
N
∣
u
n
(
x
)
−
u
n
(
y
)
∣
p
−
2
(
u
n
(
x
)
−
u
n
(
y
)
)
(
φ
(
x
)
−
φ
(
y
)
)
∣
x
−
y
∣
N
+
p
s
d
x
d
y
+
λ
∫
R
N
V
(
x
)
∣
u
n
(
x
)
∣
p
−
2
u
n
(
x
)
φ
(
x
)
d
x
−
∫
R
N
u
n
+
(
x
)
q
−
1
φ
(
x
)
d
x
→
∫
R
N
∫
R
N
∣
u
(
x
)
−
u
(
y
)
∣
p
−
2
(
u
(
x
)
−
u
(
y
)
)
(
φ
(
x
)
−
φ
(
y
)
)
∣
x
−
y
∣
N
+
p
s
d
x
d
y
+
λ
∫
R
N
V
(
x
)
u
(
x
)
p
−
1
φ
(
x
)
d
x
−
∫
R
N
u
+
(
x
)
q
−
1
φ
(
x
)
d
x
=
⟨
J
λ
′
(
u
)
,
φ
⟩
.
Therefore,
(2.16)
⟨
J
λ
′
(
u
)
,
φ
⟩
=
lim
n
→
∞
⟨
J
λ
′
(
u
n
)
,
φ
⟩
=
0
,
which implies that
u
is a critical point of
J
λ
.
Let
u
n
1
=
u
n
−
u
, we will show that as
n
→
∞
,
(2.17)
J
λ
(
u
n
1
)
→
c
−
J
λ
(
u
)
and
(2.18)
J
λ
′
(
u
n
1
)
→
0
.
To show (2.17), we observe that
(2.19)
J
λ
(
u
n
1
)
=
1
p
∫
R
N
∫
R
N
∣
u
n
1
(
x
)
−
u
n
1
(
y
)
∣
p
∣
x
−
y
∣
N
+
p
s
d
x
d
y
+
λ
p
∫
R
N
V
(
x
)
∣
u
n
1
(
x
)
∣
p
d
x
−
1
q
∫
R
N
u
n
1
+
(
x
)
q
d
x
=
J
λ
(
u
n
)
−
J
λ
(
u
)
+
λ
p
∫
R
N
V
(
x
)
(
∣
u
n
1
(
x
)
∣
p
−
∣
u
n
(
x
)
∣
p
+
∣
u
(
x
)
∣
p
)
d
x
+
1
p
∫
R
N
∫
R
N
∣
u
n
1
(
x
)
−
u
n
1
(
y
)
∣
p
−
∣
u
n
(
x
)
−
u
n
(
y
)
∣
p
+
∣
u
(
x
)
−
u
(
y
)
∣
p
∣
x
−
y
∣
N
+
p
s
d
x
d
y
+
1
q
∫
R
N
u
n
+
(
x
)
q
d
x
−
1
q
∫
R
N
u
n
1
+
(
x
)
q
d
x
−
1
q
∫
R
N
u
+
(
x
)
q
d
x
.
From Lemma 2.6,
∫
R
N
u
n
+
(
x
)
q
d
x
−
∫
R
N
u
+
(
x
)
q
d
x
−
∫
R
N
u
n
1
+
(
x
)
q
d
x
→
0
as
n
→
∞
. Conversely, we know that
‖
u
n
‖
λ
p
−
‖
u
‖
λ
p
−
‖
u
n
1
‖
λ
p
→
0
, as
n
→
∞
. Thus, from (2.19), we indeed have obtained (2.17). Now we come to show (2.18). From (2.16), we have for any
φ
∈
E
λ
⟨
J
λ
′
(
u
n
1
)
,
φ
⟩
=
⟨
J
λ
′
(
u
n
)
,
φ
⟩
−
∫
R
N
(
u
n
1
+
)
q
−
1
φ
(
x
)
d
x
+
∫
R
N
(
u
n
+
)
q
−
1
φ
(
x
)
d
x
−
∫
R
N
(
u
+
)
q
−
1
φ
(
x
)
d
x
+
o
(
1
)
.
Since
J
λ
′
(
u
n
)
→
0
and
u
n
⇀
u
in
L
q
(
R
N
)
, we have
lim
n
→
∞
sup
‖
φ
‖
λ
≤
1
∫
R
N
(
(
u
n
1
+
)
q
−
1
(
x
)
φ
(
x
)
−
(
u
n
+
)
q
−
1
φ
(
x
)
+
(
u
+
)
q
−
1
φ
(
x
)
)
d
x
=
0
.
Thus, we have
lim
n
→
∞
⟨
J
λ
′
(
u
n
1
)
,
φ
⟩
=
0
for any
φ
∈
E
λ
,
which implies (2.18), and this completes the proof.□
Proposition 2.8
Suppose
(
V
1
)
and
(
V
2
)
hold. Then, for any
C
0
>
0
, there exists
Λ
0
>
0
such that
J
λ
satisfies the
(
P
S
)
c
condition for all
λ
≥
Λ
0
and
c
≤
C
0
.
Proof
Choose
0
<
ε
<
δ
0
c
0
/
2
, where
c
0
and
δ
0
are the constants in Lemmas 2.3 and 2.4, respectively. Let
Λ
0
≔
Λ
ε
, where
Λ
ε
>
0
is from Lemma 2.5.
Assume
{
u
n
}
is a
(
P
S
)
c
sequence of
J
λ
with
λ
≥
Λ
0
and
c
≤
C
0
. By Lemma 2.7,
u
n
1
=
u
n
−
u
is a
(
P
S
)
c
′
sequence of
J
λ
with
c
′
=
c
−
J
λ
(
u
)
. If
c
′
>
0
, it follows from Lemma 2.3 that
c
′
≥
c
0
. From Lemma 2.4, we can obtain
lim inf
n
→
∞
‖
u
n
1
+
(
⋅
)
‖
L
q
(
R
N
)
q
≥
δ
0
c
′
≥
δ
0
c
0
.
Conversely, Lemma 2.5 implies
lim sup
n
→
∞
∫
B
R
ε
c
u
n
1
+
(
x
)
q
≤
ε
<
δ
0
c
0
2
.
Noting
u
n
1
→
0
in
L
loc
q
(
R
N
)
,
p
≤
q
<
p
s
∗
, a contradiction follows from the aforementioned two inequalities. Therefore,
c
′
=
0
. Thus,
u
n
1
→
0
in
E
λ
by Lemma 2.3.□
Corollary 2.9
For any
q
∈
(
p
,
p
s
∗
)
, there exists
Λ
0
>
0
, such that
c
λ
is achieved for all
λ
≥
Λ
0
at some
u
λ
∈
E
λ
, which is a ground state solution of (1.1).
Proof
By Ekeland variational principle, there is a PS sequence
u
n
∈
E
λ
, such that
J
λ
(
u
n
)
→
c
λ
and
J
λ
′
(
u
n
)
→
0
.
By Proposition 2.8, there exists some
u
λ
∈
E
λ
, such that, up to subsequence,
u
n
→
u
λ
in
E
λ
as
n
→
∞
and
λ
is sufficiently large. It is not difficult to show that
J
λ
(
u
n
)
→
J
λ
(
u
λ
)
and
J
λ
′
(
u
n
)
→
J
λ
′
(
u
λ
)
.
Therefore, we have
J
λ
(
u
λ
)
=
c
λ
and
J
λ
′
(
u
λ
)
=
0
.
This means that
u
λ
is a ground state solution of (1.1).□
4 The proof of the main results
Lemma 4.1
c
λ
→
c
(
Ω
)
as
λ
→
∞
.
Proof
From the definition of
c
λ
and
c
(
Ω
)
, we know that
c
λ
≤
c
(
Ω
)
,
λ
>
0
. Furthermore,
c
λ
is monotone increasing about the parameter
λ
>
0
.
Then, there exists a constant
k
, such that
lim
n
→
∞
c
λ
n
=
k
,
where
λ
n
→
∞
. It follows from Lemma 2.3 that
k
>
0
. By Corollary 2.9, for
n
large enough, there exists a sequence
u
n
∈
ℳ
λ
n
, such that
J
λ
n
′
(
u
n
)
=
0
and
J
λ
n
(
u
n
)
=
c
λ
n
.
If
k
<
c
(
Ω
)
, it is easy to see that
{
u
n
}
is bounded in
W
s
,
p
(
R
N
)
; thus, we can assume that
u
n
⇀
u
in
E
and
(4.1)
u
n
(
x
)
→
u
(
x
)
in
L
loc
θ
(
R
N
)
for
p
≤
θ
<
p
s
∗
.
Claim 1:
u
∣
Ω
c
=
0
. In fact, if
u
∣
Ω
c
≠
0
, then there exists a compact subset
F
⊂
Ω
c
with dist
(
F
,
Ω
)
>
0
, such that
u
∣
F
≠
0
. It follows from (4.1) that
∫
F
∣
u
n
(
x
)
∣
p
d
x
→
∫
F
∣
u
(
x
)
∣
p
d
x
>
0
.
However, there exists
ε
0
>
0
, such that
V
(
x
)
≥
ε
0
>
0
,
x
∈
F
. Thus,
J
λ
n
(
u
n
)
≥
q
−
p
p
q
λ
n
∫
F
V
(
x
)
∣
u
n
(
x
)
∣
p
d
x
≥
q
−
p
p
q
λ
n
ε
0
∫
F
∣
u
n
(
x
)
∣
p
d
x
→
∞
as
n
→
∞
,
which is a contradiction. Therefore,
u
∈
E
0
.
Claim 2:
u
n
→
u
in
L
q
(
R
N
)
for
p
<
q
<
p
s
∗
. Indeed, if not, then by the concentration-compactness lemma from the study by Loins [25], there exist
δ
>
0
,
ρ
>
0
and
x
n
∈
R
N
with
∣
x
n
∣
→
∞
, such that
lim inf
n
→
∞
∫
B
ρ
(
x
n
)
∣
u
n
(
x
)
−
u
(
x
)
∣
p
d
x
≥
δ
>
0
.
Then, we have
J
λ
n
(
u
n
)
=
q
−
p
p
q
∫
R
N
∫
R
N
∣
u
n
(
x
)
−
u
n
(
y
)
∣
p
∣
x
−
y
∣
N
+
p
s
d
x
d
y
+
q
−
p
p
q
∫
R
N
λ
n
V
(
x
)
∣
u
n
(
x
)
∣
p
d
x
≥
q
−
p
p
q
λ
n
∫
B
ρ
(
x
n
)
∩
{
x
:
V
(
x
)
≥
M
0
}
V
(
x
)
∣
u
n
(
x
)
∣
p
d
x
=
q
−
p
p
q
λ
n
∫
B
ρ
(
x
n
)
∩
{
x
:
V
(
x
)
≥
M
0
}
V
(
x
)
∣
u
n
(
x
)
−
u
(
x
)
∣
p
d
x
≥
q
−
p
p
q
λ
n
M
0
∫
B
ρ
(
x
n
)
∣
u
n
(
x
)
−
u
(
x
)
∣
p
d
x
−
M
0
∫
B
ρ
(
x
n
)
∩
{
x
:
V
(
x
)
≤
M
0
}
∣
u
n
(
x
)
∣
p
d
x
≥
q
−
p
p
q
λ
n
M
0
∫
B
ρ
(
x
n
)
∣
u
n
(
x
)
−
u
(
x
)
∣
p
d
x
−
o
(
1
)
→
∞
as
n
→
∞
,
as a contradiction. So
u
n
→
u
in
L
p
(
R
N
)
. Therefore, it is easy to see that
u
≥
0
is a solution for problem (1.3). Furthermore,
k
=
lim
n
→
∞
c
λ
n
=
lim
n
→
∞
J
λ
n
(
u
n
)
=
lim
n
→
∞
1
p
−
1
q
∫
R
N
u
n
+
(
x
)
q
d
x
=
1
p
−
1
q
∫
Ω
u
+
(
x
)
q
d
x
,
which means
u
∈
N
, and then,
k
≥
c
(
Ω
)
, a contradiction. Hence,
lim
λ
→
∞
c
λ
=
c
(
Ω
)
.□
Proof of Theorem 1.3
By Corollary 2.9, there exists
u
n
∈
ℳ
λ
n
, such that
J
λ
n
(
u
n
)
=
c
λ
n
(
λ
n
→
∞
as
n
→
∞
). It is easy to see that
{
u
n
}
is bounded in
W
s
,
p
(
R
N
)
. Then, without loss of generality,
u
n
⇀
u
in
W
s
,
p
(
R
N
)
and
u
n
→
u
in
L
loc
θ
(
R
N
)
for
p
<
θ
<
p
s
∗
.
Now we prove that
u
n
→
u
strongly in
W
s
,
p
(
R
N
)
and
u
is a ground state solution of (1.3). First, as the proof of Lemma 4.1,
u
≥
0
is a solution of problem (1.3) and
u
n
+
→
u
+
strongly in
L
q
(
R
N
)
.
Now we claim that
λ
n
∫
R
N
V
(
x
)
∣
u
n
(
x
)
∣
p
d
x
→
0
and
∫
R
N
∫
R
N
∣
u
n
(
x
)
−
u
n
(
y
)
∣
p
∣
x
−
y
∣
N
+
p
s
d
x
d
y
→
∫
R
N
∫
R
N
∣
u
(
x
)
−
u
(
y
)
∣
p
∣
x
−
y
∣
N
+
p
s
d
x
d
y
.
Indeed, if either
lim inf
n
→
∞
λ
n
∫
R
N
V
(
x
)
∣
u
n
(
x
)
∣
p
d
x
>
0
or
lim inf
n
→
∞
∫
R
N
∫
R
N
∣
u
n
(
x
)
−
u
n
(
y
)
∣
p
∣
x
−
y
∣
N
+
p
s
d
x
d
y
>
∫
R
N
∫
R
N
∣
u
(
x
)
−
u
(
y
)
∣
p
∣
x
−
y
∣
N
+
p
s
d
x
d
y
.
Thus, we have
∫
R
N
∫
R
N
∣
u
(
x
)
−
u
(
y
)
∣
p
∣
x
−
y
∣
N
+
p
s
d
x
d
y
<
∫
Ω
u
+
(
x
)
q
d
x
.
Therefore, there is
α
∈
(
0
,
1
)
, such that
α
u
∈
N
and
c
(
Ω
)
