1 Introduction and Main Results
We consider the following Schrödinger equations involving fractional Laplacian and indefinite potentials:
(${\mathcal{P}_{\lambda}}$)
(
-
Δ
)
s
u
+
(
λ
a
(
x
)
-
δ
)
u
=
|
u
|
p
-
2
u
,
x
∈
ℝ
N
,
where 0<s<1, N>2s, 2<p≤2s*:=2NN-2s, a(x) is a nonnegative continuous function and the zero set a-1(0):={x∈ℝN:a(x)=0} has a nonempty, smooth and connected interior part inta-1(0).
The fractional Laplacian has been studied extensively in recent years. In [9], Caffarelli and Silvestre showed that the fractional Laplacian (-Δ)s on ℝd can be expressed as the Dirichlet-to-Neumann operator for a suitable local problem on the upper half space ℝ+d+1. For one-dimensional case, Frank and Lenzmann [16] proved the uniqueness and nondegeneracy of ground states for a class of semilinear fractional Laplacian equations with subcritical growth. The authors also obtained similar results in [17] for the higher-dimensional case. Via a Lyapunov–Schmidt reduction method, Dávila, del Pino and Wei [13] proved the existence and concentration of standing waves for a class of semilinear fractional Laplacian equations. For more details of fractional Laplacian, please see [3, 7, 8, 10, 11, 19, 28, 30] and the references therein.
Many papers deal with Schrödinger equations involving Laplacian and potential wells. For λ large enough, the operator -Δ+λa(x)-δ is positive definite if δ<λ1, where λ1 is the principle eigenvalue for -Δ on functions defined in inta-1(0). If δ>λ1, the operator -Δ+λa(x)-δ is indefinite. Take δ=-1, the positive definite case, Bartsch and Wang [6] considered (${\mathcal{P}_{\lambda}}$) with a more general nonlinearity f(x,u) satisfies some conditions. The authors proved that there exists Λ1>0 such that (${\mathcal{P}_{\lambda}}$) has a positive and a negative solution for λ>Λ1. Furthermore, if f(x,u) is odd in u, the authors also proved that for any k∈ℕ there exists Λk>0 such that (${\mathcal{P}_{\lambda}}$) has at least k pairs ±u1,±u2,…,±uk of nontrivial solutions for λ≥Λk. For more details about the positive definite case, please see [1, 2, 4, 21] and the references therein.
For the indefinite case, Ding and Wei [15] considered the following problem:
(1.1)
{
-
Δ
u
(
x
)
+
λ
V
(
x
)
u
(
x
)
=
λ
|
u
(
x
)
|
p
-
2
u
(
x
)
+
λ
g
(
x
,
u
)
,
x
∈
ℝ
N
,
u
(
x
)
→
0
as
|
x
|
→
∞
,
where V(x) can be negative in some domains in ℝN, g(x,u) is a perturbation term. Both for subcritical growth and critical growth, using variational methods, the authors proved that there exists Λ>0 such that for any λ>Λ, problem (1.1) admits at least one nontrivial solution. In [29], Szulkin and Weth gave a new minimax characterization of the corresponding critical value and hence reduced the indefinite problem to a definite one. The authors also presented a precise description of the Nehari–Pankov manifold which is useful even for other problems. Using this type of manifold, Bartsch and the first author [5] proved the existence of multi-bump solutions for semilinear equations with indefinite potentials. For more details about indefinite case, please see [14, 18] and the references therein.
In present paper, we are interested in the existence and concentration phenomenon of least energy solutions for fractional Laplacian with indefinite potentials. Motivated by Servadei and Valdinoci [24, 26, 27, 25], we consider our problem directly in the corresponding fractional Sobolev space without taking s-harmonic extension. We will prove that (${\mathcal{P}_{\lambda}}$) has a least energy solution which localizes near the potential well and changes sign for λ large. Our assumptions about a(x) and δ are as follows.
a
(
x
)
∈
C
(
ℝ
N
,
ℝ
)
satisfies a(x)≥0 and Ω:=int{x∈ℝN:a(x)=0} is nonempty, the boundary of Ω is smooth and Ω¯={x∈ℝN:a(x)=0}.
a
0
=
1
2
lim inf
|
x
|
→
∞
a
(
x
)
>
0
.
The operator (-Δ)s-δ defined in inta-1(0) is indefinite and nondegenerate. Namely, λk<δ<λk+1 for some k≥1. Here {λk} is the class of all eigenvalues for (-Δ)s on functions defined in inta-1(0).
Remark 1.1.
Assume (A1) and (A2) hold; we see that Ω is bounded in ℝN. In fact, there exists R>0 such that
(1.2)
Ω
⊆
B
R
(
0
)
,
a
(
x
)
>
a
0
for all
|
x
|
>
R
,
where BR(0) is the ball centered at 0 with radii R. Of course, we can replace (A2) by the following one.
There exists a constant M>0 such that the Lebesgue measure of the set {x∈ℝN:a(x)<M} is finite, i.e. |{x∈ℝN:a(x)<M}|<+∞.
Let Hs(ℝN) be the standard fractional Sobolev space defined by
H
s
(
ℝ
N
)
:=
{
u
:
∫
ℝ
N
∫
ℝ
N
|
u
(
x
)
-
u
(
y
)
|
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
<
+
∞
,
∫
ℝ
N
u
2
𝑑
x
<
+
∞
}
with the inner product
〈
u
,
v
〉
:=
∫
ℝ
N
∫
ℝ
N
(
u
(
x
)
-
u
(
y
)
)
(
v
(
x
)
-
v
(
y
)
)
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
+
∫
ℝ
N
u
v
𝑑
x
,
and the induced norm is
∥
u
∥
:=
(
∫
ℝ
N
∫
ℝ
N
|
u
(
x
)
-
u
(
y
)
|
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
+
∫
ℝ
N
u
2
𝑑
x
)
1
2
.
Let
E
0
:=
{
u
∈
H
s
(
ℝ
N
)
:
u
=
0
a.e. in
ℝ
N
∖
Ω
}
be a subspace of Hs(ℝN) endowed with a new norm
∥
u
∥
0
=
(
∫
ℝ
N
∫
ℝ
N
|
u
(
x
)
-
u
(
y
)
|
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
)
1
2
,
u
∈
E
0
.
As described in Section 2.2, ∥⋅∥0 and ∥⋅∥ are equivalent and (E0,∥⋅∥0) is a Hilbert space.
Denote
V
λ
=
λ
a
(
x
)
-
δ
,
V
λ
+
=
max
{
0
,
V
λ
}
,
V
λ
-
=
max
{
0
,
-
V
λ
}
.
Let
E
λ
:=
{
u
∈
H
s
(
ℝ
N
)
:
∫
ℝ
N
a
(
x
)
u
2
𝑑
x
<
∞
}
be the Hilbert space endowed with the norm
∥
u
∥
λ
:=
(
∫
ℝ
N
∫
ℝ
N
|
u
(
x
)
-
u
(
y
)
|
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
+
∫
ℝ
N
V
λ
+
u
2
𝑑
x
)
1
2
for all
u
∈
E
λ
.
The energy functional corresponding to (${\mathcal{P}_{\lambda}}$) is
I
λ
(
u
)
:=
1
2
∫
ℝ
N
∫
ℝ
N
|
u
(
x
)
-
u
(
y
)
|
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
-
1
2
∫
ℝ
N
V
λ
u
2
𝑑
x
-
1
p
∫
ℝ
N
|
u
|
p
𝑑
x
for all
u
∈
E
λ
.
with Fréchet derivative
I
λ
′
(
u
)
v
:=
∫
ℝ
N
∫
ℝ
N
(
u
(
x
)
-
u
(
y
)
)
(
v
(
x
)
-
v
(
y
)
)
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
+
∫
ℝ
N
V
λ
u
v
𝑑
x
-
∫
ℝ
N
|
u
|
p
-
2
u
v
𝑑
x
for all
v
∈
E
λ
.
We say uλ∈Eλ is a solution to (${\mathcal{P}_{\lambda}}$) if uλ∈Eλ is a critical point of Iλ, i.e. Iλ′(uλ)=0.
Let us denote Lλ:=(-Δ)s+λa(x)-δ and L0:=(-Δ)s-δ. Take {μk(L0)} be the class of all eigenvalues of L0 in E0(each eigenvalue is repeated according to its multiplicity). As proved in Section 2.3, if μ1(L0)>0, then (-Δ)s+λa(x)-δ is positive definite and the ground state of (${\mathcal{P}_{\lambda}}$) is positive for λ large enough. If μ1(L0)<0 and μk(L0)≠0 for any k, then (-Δ)s+λa(x)-δ is indefinite and the least energy solutions of (${\mathcal{P}_{\lambda}}$) are sign changing for λ large enough.
Let ek be an eigenfunction corresponding to μk(L0) for k=1,2,3,…. We may assume that {ek}k≥1 is an orthogonal base of E0 and L2(Ω) (see Lemma 2.3 in Section 2.3). By (A3), E0 can be split as an orthogonal sum E0-⊕E0+ according to the positive and negative eigenvalues of L0, i.e.
E
0
=
E
0
-
⊕
E
0
+
,
where
E
0
-
=
span
{
e
1
,
e
2
,
…
,
e
k
}
,
E
0
+
=
{
u
∈
E
0
:
∫
Ω
u
v
𝑑
x
=
0
for all
v
∈
E
0
-
}
.
As proved in Section 2.3, we see that infσe(Lλ)≥λa0-δ, where σess(Lλ) is the essential spectrum of Lλ. Moreover, there exist a finite number of k such that μk(Lλ)<0, i.e. Lλ thus has finite Morse index. For large λ fixed, Lλ has finite eigenvalues below σess(Lλ). By the study of asymptotic behavior of eigenvalues for Lλ as λ→+∞, we see that 0 is not an eigenvalue of Lλ for λ large. Therefore, Eλ can be split as an orthogonal sum Eλ=Eλ-⊕Eλ+ according to the negative and positive eigenvalues of Lλ.
Note that for λ large enough, the space Eλ- are close to E0- (please see Section 2.3). In order to study the asymptotic behavior of least energy solutions for (${\mathcal{P}_{\lambda}}$), we need to modify the Nehari–Pankov manifold. To do that, let Pλ-:Eλ↦E0- and P0-:E0↦E0- be two orthogonal projections. Define the modified Nehari–Pankov manifold of Iλ by
𝒩
λ
:=
{
u
∈
E
λ
∖
{
0
}
:
P
λ
-
∇
I
λ
(
u
)
=
0
,
I
λ
′
(
u
)
u
=
0
}
⊂
E
λ
∖
E
0
-
,
with the corresponding level
c
λ
:=
inf
u
∈
𝒩
λ
I
λ
(
u
)
.
As λ→+∞, the limit problem of (${\mathcal{P}_{\lambda}}$) is
(1.3)
{
(
-
Δ
)
s
u
-
δ
u
=
|
u
|
p
-
2
u
in
Ω
,
u
=
0
in
ℝ
N
∖
Ω
with the energy functional
I
0
(
u
)
=
1
2
∫
ℝ
N
∫
ℝ
N
|
u
(
x
)
-
u
(
y
)
|
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
-
1
2
∫
Ω
δ
u
2
𝑑
x
-
1
p
∫
Ω
|
u
|
p
𝑑
x
for all
u
∈
E
0
.
The Nehari–Pankov manifold of I0(u) is defined by
𝒩
0
=
{
u
∈
E
0
∖
{
0
}
:
P
0
-
∇
I
0
(
u
)
=
0
,
I
0
′
(
u
)
u
=
0
}
with the corresponding level
c
0
:=
inf
𝒩
0
I
0
(
u
)
,
where I0′(u) is the Fréchet derivative of I0(u).
Remark 1.2.
The Nehari–Pankov manifold was firstly introduced by Pankov [22]. This type of manifold coincides with the Nehari manifold if all eigenvalues are positive, i.e. the negative eigenfunction space is {0}. Moreover, the minimizer for c0 is a least energy solution of problem (1.3). By the definition of the modified Nehari–Pankov manifold 𝒩λ, it is easy to see that Pλ-u=P0-u for any u∈E0. As proved in Lemma 2.9, we see that the minimizer for cλ is indeed a least energy solution of (${\mathcal{P}_{\lambda}}$), the solution with smallest energy among all nontrivial solutions.
Our main results are as follows.
Theorem 1.3 (Existence).
Assume (A1), (A2), (A3) and one of the following conditions hold:
N
>
2
s
, p is subcritical, i.e. 2<p≤2s*,
N
≥
4
s
, p is critical, i.e. p=2s*.
Then there exists Λ>0 such that for any λ≥Λ, (Pλ) admits a least energy solution uλ∈Eλ which achieves cλ.
Theorem 1.4 (Asymptotic Behavior).
Under the assumptions of Theorem 1.3, let uλ be a least energy solution to (Pλ) for λ≥Λ. Then up to a subsequence, for any sequence λn→+∞, the sequence {uλn} converges to a least energy solution of the limit problem (1.3) in Hs(RN).
The following is an extension of Theorem 1.4.
Theorem 1.5.
Under the assumptions of Theorem 1.3, let uλ be a nontrivial solution to (Pλ) for λ≥Λ. Suppose that one of the following conditions holds:
lim sup
λ
→
+
∞
I
λ
(
u
λ
)
<
+
∞
for
2
<
p
<
2
s
*
,
lim sup
λ
→
+
∞
I
λ
(
u
λ
)
<
s
N
S
N
2
for
p
=
2
s
*
, where
S
:=
inf
u
∈
H
s
(
ℝ
N
)
,
∥
u
∥
L
2
s
*
(
ℝ
N
)
=
1
∫
ℝ
N
|
u
(
x
)
-
u
(
y
)
|
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
.
Then up to a subsequence, for any sequence λn→+∞, the sequence {uλn} converges to a solution of the limit problem (1.3) in Hs(RN).
Recall that (-Δ)s is a nonlocal operator, please see the definition of (-Δ)s in Section 2.1. Compared with local operators, such as the Laplacian -Δ, nonlocal operators seem much more difficult to deal with. There are some different phenomena between the local and nonlocal operators. For instance:
Remark 1.6.
Assume Ω:=int{x∈ℝN:a(x)=0}=Ω1∪Ω2,Ω1∩Ω2≠∅. Here Ω1 and Ω2 are two smooth domains in ℝN. We see that the least energy solutions to (𝒫λ) may localize near Ω1 and Ω2 simultaneously for λ large enough (see Lemma 3.6). This phenomena cannot happen for local operators such as the Laplacian.
In the following sections, without especially stated, we always assume λk<δ<λk+1 for some k≥1, i.e. μk(L0)<0 and μk+1(L0)>0. Our paper is organized as follows. In Section 2, we give some preliminary results. More precisely, in Section 2.1, we introduce some definitions of fractional Laplacian. In Section 2.2, we introduce some fractional Sobolev spaces. The spectrum of Lλ and L0 is studied in Section 2.3 and some properties of the Modified Nehari–Pankov manifold are proved in Section 2.4. In Section 3, we prove the existence of least energy solutions for the limit problem (1.3). Section 4 is dedicated to the proof of our main results in the subcritical case and Section 5 focus on the proof of the main results in the critical case.
2 Preliminary Results
In this section, for the convenience of the readers, we present some definitions and notations. Firstly, we introduce some definitions of the fractional Laplacian (-Δ)s. Secondly, we consider some fractional Sobolev spaces, such as Eλ and E0. Then we discuss the spectrum of Lλ and L0. At last, we give some properties of the modified Nehari–Pankov manifold.
2.1 Introduction to Fractional Laplacian Operators
Let ℑ(ℝN):={ϕ∈C0∞(ℝN):supx∈ℝN(1+|x|2)k2|Dαϕ(x)|<+∞,k,|α|=0,1,2,…} be the so-called Schwartz space with the seminorms
p
n
(
ϕ
)
=
sup
x
∈
ℝ
N
∑
|
α
|
≤
n
|
D
α
ϕ
(
x
)
|
,
n
=
1
,
2
,
…
.
We denote the topological dual of ℑ(ℝN) by ℑ′(ℝN). As usual, for any ϕ∈ℑ, we denote by
𝔉
ϕ
(
ξ
)
=
1
(
2
π
)
N
2
∫
ℝ
N
e
-
i
ξ
⋅
x
ϕ
(
x
)
𝑑
x
the Fourier transform of ϕ. For any ϕ∈ℑ, 0<s<1, we define the fractional Laplacian operator (-Δ)s as
(2.1)
(
-
Δ
)
s
ϕ
(
x
)
=
C
N
,
s
P.V.
∫
ℝ
N
ϕ
(
x
)
-
ϕ
(
y
)
|
x
-
y
|
N
+
2
s
𝑑
y
=
C
N
,
s
lim
ϵ
→
0
∫
ℝ
N
∖
B
ϵ
(
x
)
ϕ
(
x
)
-
ϕ
(
y
)
|
x
-
y
|
N
+
2
s
𝑑
y
.
Here P.V. is a commonly used abbreviation for “in the principle value sense” and CN,s is a dimensional constant that depends on N and s, precisely given by
C
N
,
s
=
(
∫
ℝ
N
1
-
cos
(
ξ
1
)
|
ξ
|
N
+
2
s
𝑑
ξ
)
-
1
.
According to (2.1), it is easy to see that (-Δ)s is a nonlocal operator. Due to the singularity of the kernel, the right-hand side of (2.1) is not well defined in general. In the case 0<s<12 the integral (2.1) is not really singular near x, see [20]. In order to get rid of P.V. in (2.1), one can redefine the fractional Laplacian operator (-Δ)s as
(2.2)
(
-
Δ
)
s
ϕ
(
x
)
=
-
C
N
,
s
2
∫
ℝ
N
u
(
x
+
y
)
+
u
(
x
-
y
)
-
2
u
(
x
)
|
y
|
N
+
2
s
𝑑
y
for all
ϕ
∈
ℑ
,
x
∈
ℝ
N
.
Now we take into account an alternative definition of the space Hs(ℝN) via the Fourier transform. Precisely, we define
H
^
s
:=
{
u
∈
L
2
(
ℝ
N
)
:
∫
ℝ
N
(
1
+
|
ξ
|
2
s
)
|
𝔉
ϕ
(
ξ
)
|
2
𝑑
ξ
<
+
∞
}
.
The fractional Laplacian (-Δ)s can be viewed as a pseudo-differential operator of symbol |ξ|2s, namely for 0<s<1,
(2.3)
(
-
Δ
)
s
ϕ
(
x
)
=
𝔉
-
1
(
|
ξ
|
2
s
(
𝔉
u
)
)
for all
u
∈
ℑ
,
ξ
∈
ℝ
N
.
As proved in [20], Hs(ℝN) and H^s are coincide and the above definitions of (-Δ)s are also coincide.
Recently Caffarelli and Silvestre [7] considered the following boundary value problem in the half space ℝ+N+1:
{
∇
⋅
(
t
1
-
2
s
∇
ϕ
~
)
=
0
in
ℝ
+
N
+
1
,
ϕ
~
(
x
,
0
)
=
ϕ
(
x
)
in
ℝ
N
.
Here ϕ~(x,t) is the so-called s-harmonic extension of ϕ, explicitly given as a convolution integral with the s-Poisson kernel ps(x,y),
ϕ
~
(
x
,
t
)
=
∫
ℝ
N
p
s
(
x
-
z
,
t
)
ϕ
(
z
)
𝑑
z
,
where
p
s
(
x
,
t
)
=
d
N
,
s
t
4
s
-
1
(
|
x
|
2
+
|
t
|
2
)
N
-
1
+
4
s
2
and dN,s achieves ∫ℝNps(x,t)𝑑x=1. Then under suitable regularity, (-Δ)s is the Dirichlet-to-Neumann map for this problem, namely
(2.4)
(
-
Δ
)
s
ϕ
(
x
)
=
-
lim
t
→
0
+
t
1
-
2
s
∂
t
ϕ
~
(
x
,
t
)
.
Thus by this new transformation, the nonlocal operator (-Δ)s may be reduced to a local, possibly singular or degenerate operator on functions sitting in the higher-dimensional half space ℝ+N+1=ℝN×(0,+∞).
The definitions (2.1), (2.2), (2.3) and (2.4) of the fractional Laplacian (-Δ)s are all equivalent for instance in Schwartz’s space of rapidly decreasing smooth functions. For more details about the fractional Laplacian, one refers to [9, 20] and the references therein.
In the present paper, we use (2.1) as the definition of (-Δ)s and for the simplicity and without loss of generality, we replace the constant CN,s by 1 throughout this paper.
2.2 Some Fractional Sobolev Spaces
As we know, Hs(ℝN) can be continuously embedded into Lp(ℝN) for 2≤p≤2s*, and compactly embedded into Llocp(ℝN) for 2≤p<2s*. In particular, the following fractional Sobolev inequality holds:
(2.5)
S
=
inf
{
∫
ℝ
N
∫
ℝ
N
|
u
(
x
)
-
u
(
y
)
|
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
:
u
∈
H
s
(
ℝ
N
)
∖
{
0
}
,
∥
u
∥
L
2
s
*
(
ℝ
N
)
=
1
}
>
0
.
In [12], Cotsiolis and Tavoularis showed that S is achieved by
v
=
V
(
x
)
∥
V
∥
L
2
s
*
(
ℝ
N
)
,
where
V
(
x
)
=
κ
(
μ
2
+
|
x
-
x
0
|
2
)
-
N
-
2
s
2
for κ∈ℝ∖{0}, μ∈ℝ and x0∈ℝN fixed.
Let Ω⊆ℝN be a bounded smooth domain in ℝN, and let Hs(Ω) be the standard fractional Sobolev space. The embedding Hs(Ω)↪Lp(Ω) is continuous for 1≤p≤2s*, and compact for 1≤p<2s*. For more details about the fractional Sobolev spaces, one refers to [20] and the references therein.
Recall that E0={u∈Hs(ℝN):u=0 a.e. in ℝN∖Ω}. In [12], Cotsiolis and Tavoularis gave a sort of Poincaré–Sobolev inequality as follows:
∫
Ω
|
u
|
2
𝑑
x
≤
C
∫
ℝ
N
∫
ℝ
N
|
u
(
x
)
-
u
(
y
)
|
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
for all
u
∈
E
0
,
where C>0 depends on Ω and N. According to the definitions of the norms ∥⋅∥0 and ∥⋅∥, we see that the two norms are equivalent in E0. Moreover, (E0,∥⋅∥0) is also a Hilbert space (see [12, Lemma 7]). The embedding E0↪Lp(Ω) is continuous for 1≤p≤2s*, and compact for 1≤p<2s* (see [12, Lemma 8]).
We finish this subsection by giving the following uniformly imbedding lemma.
Lemma 2.1.
There exists a constant λ0>0 such that the embedding Eλ↪Hs(RN) is continuous, uniformly in λ for λ>λ0.
Proof.
By (1.2) in Remark 1.1, Ω⊆BR(0) for large R>0 and a(x)>a0 for each x∈ℝN∖BR(0). Let ξ:ℝN→ℝ be a smooth cutoff function satisfying
(2.6)
0
≤
ξ
≤
1
,
ξ
=
1
in
B
R
(
0
)
,
ξ
=
0
in
ℝ
N
∖
B
R
+
1
(
0
)
.
Take λ0=a0+δa0; then by (2.6), for λ≥λ0, we have
(2.7)
∫
ℝ
N
(
1
-
ξ
2
)
u
2
𝑑
x
=
∫
ℝ
N
∖
B
R
(
0
)
(
1
-
ξ
2
)
u
2
𝑑
x
(2.8)
≤
1
a
0
∫
ℝ
N
∖
B
R
(
0
)
(
λ
a
(
x
)
-
δ
)
u
2
𝑑
x
=
1
a
0
∫
ℝ
N
∖
B
R
(
0
)
V
λ
+
u
2
𝑑
x
(2.9)
≤
1
a
0
∫
ℝ
N
V
λ
+
u
2
𝑑
x
.
By (2.6), Hölder’s inequality and the definition of S, we find that
(2.10)
∫
ℝ
N
ξ
2
u
2
𝑑
x
=
∫
B
R
+
1
(
0
)
ξ
2
u
2
𝑑
x
(2.11)
≤
|
B
R
+
1
(
0
)
|
1
-
2
2
s
*
(
∫
ℝ
N
|
u
|
2
s
*
𝑑
x
)
2
2
s
*
(2.12)
≤
|
B
R
+
1
(
0
)
|
1
-
2
2
s
*
S
∫
ℝ
N
∫
ℝ
N
|
u
(
x
)
-
u
(
y
)
|
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
.
According to (2.9), (2.12) and the definition of the norm ∥⋅∥, we obtain that
∥
u
∥
2
=
∫
ℝ
N
∫
ℝ
N
|
u
(
x
)
-
u
(
y
)
|
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
+
∫
ℝ
N
u
2
𝑑
x
≤
(
|
B
R
+
1
(
0
)
|
1
-
2
2
s
*
+
1
)
∫
ℝ
N
∫
ℝ
N
|
u
(
x
)
-
u
(
y
)
|
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
+
1
a
0
∫
ℝ
N
V
λ
+
u
2
𝑑
x
≤
C
(
∫
ℝ
N
∫
ℝ
N
|
u
(
x
)
-
u
(
y
)
|
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
+
∫
ℝ
N
V
λ
+
u
2
𝑑
x
)
≤
C
∥
u
∥
λ
2
.
Thus ∥u∥≤C∥u∥λ, where C>0 is independent of λ. ∎
Remark 2.2.
For λ>λ0, since the embedding constant for Eλ↪Hs(ℝN) does not dependent on λ, Eλ keeps all properties of Hs(ℝN). Precisely, for λ>λ0, Eλ can be continuously embedded into Lp(ℝN) for 2≤p≤2s* and compactly into Llocp(ℝN) for 2≤p<2s*. Moreover, all embedding constants do not depend on λ.
2.3 Eigenvalue Problems with Fractional Laplacian Operators
Firstly, we consider the spectrum of the operator L0 in E0. Recall that Ω=inta-1(0) is a smooth bounded domain in ℝN. For 0<s<1, we say λ is an eigenvalue of (-Δ)s in E0 if there exists u∈E0 such that
(2.13)
∫
ℝ
N
∫
ℝ
N
(
u
(
x
)
-
u
(
y
)
)
(
v
(
x
)
-
v
(
y
)
)
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
=
λ
∫
Ω
u
(
x
)
v
(
x
)
𝑑
x
for any v∈E0, i.e. (u,λ) is a weak solution to the equation
(2.14)
{
(
-
Δ
)
s
u
(
x
)
=
λ
u
(
x
)
in
Ω
,
u
(
x
)
=
0
in
ℝ
N
∖
Ω
,
A function u∈E0 satisfying (2.13) is called an eigenfunction corresponding to λ. For convenience, we also say λ is an eigenvalue of (2.14).
The following lemma described the eigenvalues and eigenfunctions to (2.14).
Lemma 2.3 ([26, Proposition 9]).
Let 0<s<1, N>2s, and let Ω be an open bounded set of RN.
Problem (
2.14
) admits an eigenvalue
λ
1
which is positive and that can be characterized as
(2.15)
λ
1
=
min
{
∫
ℝ
N
∫
ℝ
N
|
u
(
x
)
-
u
(
y
)
|
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
:
u
∈
E
0
,
∥
u
∥
L
2
(
Ω
)
=
1
}
.
There exists a nonnegative function
e
1
∈
E
0
, which is an eigenfunction corresponding to
λ
1
, attaining the minimum in (
2.15
), that is
∥
e
1
∥
L
2
(
Ω
)
=
1
and
λ
1
=
∫
ℝ
N
∫
ℝ
N
|
e
1
(
x
)
-
e
1
(
y
)
|
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
.
λ
1
is simple, that is if
u
∈
E
0
is a solution to the equation
∫
ℝ
N
∫
ℝ
N
(
u
(
x
)
-
u
(
y
)
)
(
ϕ
(
x
)
-
ϕ
(
y
)
)
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
=
λ
1
∫
Ω
u
(
x
)
ϕ
(
x
)
𝑑
x
,
u
∈
E
0
,
for all
ϕ
(
x
)
∈
E
0
,
then
u
=
k
e
1
, with
k
∈
ℝ
.
The set of the eigenvalues of problem (
2.14
) consists of a sequence
{
λ
k
}
k
≥
1
with
0
<
λ
1
<
λ
2
≤
⋯
≤
λ
k
≤
λ
k
+
1
≤
⋯
and
λ
k
→
+
∞
as
k
→
+
∞
.
Moreover, for any
k
≥
1
, the eigenvalues can be characterized as
(2.16)
λ
k
+
1
=
min
{
∫
ℝ
2
N
|
u
(
x
)
-
u
(
y
)
|
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
:
u
∈
𝒫
k
+
1
,
∥
u
∥
L
2
(
Ω
)
=
1
}
,
where
𝒫
k
+
1
=
{
u
∈
E
0
:
∫
ℝ
2
N
(
u
(
x
)
-
u
(
y
)
)
(
e
j
(
x
)
-
e
j
(
y
)
)
|
x
-
y
|
N
+
2
s
=
0
,
j
=
1
,
…
,
k
}
.
For any
k
≥
1
, there exists a function
e
k
+
1
, which is an eigenfunction corresponding to
λ
k
+
1
, attaining the minimum in (
2.16
), that is
∥
e
k
+
1
∥
L
2
(
Ω
)
=
1
and
λ
k
+
1
=
∫
ℝ
N
∫
ℝ
N
|
e
k
+
1
(
x
)
-
e
k
+
1
(
y
)
|
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
.
The sequence
{
e
k
}
k
≥
1
of eigenfunctions corresponding to
λ
k
is an orthonormal basis of
L
2
(
Ω
)
and an orthogonal basis of
E
0
.
Each eigenvalue
λ
k
has finite multiplicity; more precisely, if
λ
k
is such that
λ
k
-
1
<
λ
k
=
λ
k
+
1
=
⋯
=
λ
k
+
h
<
λ
k
+
h
+
1
for some
h
≥
0
, then the set of all eigenfunctions corresponding to
λ
k
agrees with
span
{
e
k
,
e
k
+
1
,
…
,
e
k
+
h
}
.
Next, we study the spectrum of Lλ in Eλ. It is easy to see that Lλ is self-adjoint in L2(ℝN) and bounded from below by -δ. Denote
U
λ
(
ψ
1
,
ψ
2
,
…
,
ψ
m
)
=
inf
{
L
λ
(
ψ
)
ψ
:
ψ
∈
E
λ
,
∥
ψ
∥
L
2
(
ℝ
N
)
=
1
,
(
ψ
,
ψ
j
)
=
0
,
j
=
1
,
2
,
…
,
m
}
,
where
L
λ
(
ψ
)
ψ
:=
∫
ℝ
N
∫
ℝ
N
|
ψ
(
x
)
-
ψ
(
y
)
|
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
+
∫
ℝ
N
V
λ
ψ
2
𝑑
x
,
∥
ψ
∥
L
2
(
ℝ
N
)
2
:=
∫
ℝ
N
ψ
2
𝑑
x
,
(
ψ
,
ψ
j
)
:=
∫
ℝ
N
ψ
ψ
j
𝑑
x
=
0
.
For k∈ℕ fixed, we define spectral values of Lλ by the k-th Rayleigh quotient
μ
k
(
L
λ
)
:=
sup
{
U
λ
(
ψ
1
,
ψ
2
,
…
,
ψ
k
-
1
)
:
ψ
1
,
ψ
2
,
…
,
ψ
k
-
1
∈
E
λ
}
.
Obviously, μk(Lλ) is nondecreasing respect to k and λ. By [23, Theorem XIII.1 and Theorem XIII.2], we know that either μk(Lλ) is an eigenvalue of Lλ or μk(Lλ)=μk+1(Lλ)=⋯=infσess(Lλ). Denote μk(L0):=λk-δ. By Lemma 2.3, it is easy to see that {μk(L0)}k≥1 is the class of all eigenvalues of L0:=(-Δ)s-δ in E0. Moreover, μk(L0) can be characterized as
μ
k
(
L
0
)
=
max
S
∈
∑
k
-
1
min
ψ
∈
S
⊥
,
∥
ψ
∥
L
2
(
Ω
)
=
1
{
L
0
(
ψ
)
ψ
}
,
where
L
0
(
ψ
)
ψ
:=
∫
ℝ
N
∫
ℝ
N
|
ψ
(
x
)
-
ψ
(
y
)
|
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
-
δ
∫
Ω
ψ
2
𝑑
x
,
and ∑k-1 denotes the collection of (k-1)-dimensional subspaces of E0. Since Lλ(ψ)ψ=L0(ψ)ψ for each ψ∈E0, we have μk(Lλ)≤μk(L0). Thus in order to prove the existence of eigenvalues of Lλ in Eλ, we just need to prove that infσess(Lλ)→+∞ as λ→+∞.
Lemma 2.4.
Under the assumptions (A1)–(A3), the essential spectrum σess(Lλ) is contained in [λa0-δ,+∞) for λ>λ0. Furthermore, infσess(Lλ)→+∞ as λ→+∞.
Proof.
The proof of this lemma is similar to [4, Proposition 2.3]. For the convenience of the reader, we give the details.
We set Wλ=Vλ-λa0+δ=λ(a(x)-a0) and write Wλ1=max{Wλ,0} and Wλ2=min{Wλ,0}. Obviously, for λ>λ0,
(2.17)
σ
(
(
-
Δ
)
s
+
W
λ
1
+
λ
a
0
-
δ
)
⊂
[
λ
a
0
-
δ
,
+
∞
)
for Wλ1≥0. Let Hλ:=(-Δ)s+Wλ1+λa0-δ; then Lλ=Hλ+Wλ2.
We claim that Wλ2 is a relative form compact perturbation of Lλ. Since Wλ2 is bounded, the form domain of Hλ is the same as the form domain Eλ of Lλ. Thus we have to show that Eλ↦Eλ*, u↦Wλ2⋅u is compact. Select a bounded sequence {un}n≥1 in 〈Eλ,∥⋅∥λ〉; then according to Lemma 2.1, {un}n≥1 is also a bounded sequence in Hs(ℝN). Thus there exists a function u∈Hs(ℝN) such that, up to a subsequence,
(2.18)
{
u
n
⇀
u
weakly in
H
s
(
ℝ
N
)
,
u
n
→
u
strongly in
L
loc
2
(
ℝ
N
)
,
u
n
→
u
a.e. in
ℝ
N
as n→+∞. Notice that a(x)>a0 for each x∈ℝN∖BR (see (1.2) in Remark 1.1), then the support set of Wλ2 is contained in BR. Thus by Hölder’s inequality, and Lemma 2.1, for λ>λ0, we have
|
∫
ℝ
N
W
λ
2
(
u
n
-
u
)
v
𝑑
x
|
=
|
∫
B
R
W
λ
2
(
u
n
-
u
)
v
𝑑
x
|
≤
δ
∫
B
R
|
(
u
n
-
u
)
v
|
𝑑
x
≤
δ
(
∫
B
R
(
u
n
-
u
)
2
𝑑
x
)
1
2
∥
v
∥
≤
C
(
∫
B
R
(
u
n
-
u
)
2
𝑑
x
)
1
2
∥
v
∥
λ
.
Hence by (2.18), we have
∥
W
λ
2
u
n
-
W
λ
2
u
∥
E
λ
*
≤
C
(
∫
B
R
(
u
n
-
u
)
2
𝑑
x
)
1
2
→
0
as n→+∞. Thus Wλ2 is a relative form compact perturbation of Lλ.
According to the Classical Weyl Theorem (see [23, Example 3, p. 117]), σess(Lλ)=σess(Hλ). Thus by (2.17), for λ>λ0, σess(Lλ)⊂[λa0-δ,+∞). Moreover, infσess(Lλ)→+∞ as λ→+∞. ∎
At last, we deal with the asymptotic behavior of the eigenvalues of Lλ in Eλ. Let {μkλ}k≥1 be the class of all distinct eigenvalues of Lλ:=(-Δ)s+λa(x)-δ in Eλ, and let {μk}k≥1 be the class of all distinct eigenvalues of L0:=(-Δ)s-δ in E0. Without loss of generality, we can order these eigenvalues as follows:
μ
1
λ
<
μ
2
λ
<
μ
3
λ
<
…
<
μ
k
λ
λ
<
inf
σ
ess
(
L
λ
)
,
and
μ
1
<
μ
2
<
μ
3
<
…
<
μ
k
<
0
<
μ
k
+
1
<
⋯
.
Moreover, μkλλ→+∞ as λ→+∞, and μk→+∞ as k→+∞. Let Vjλ be the eigenfunction space corresponding to μjλ and let Vj be the eigenfunction space corresponding to μj. We say Vkλ converges to Vk, i.e. Vkλ→Vk as k→+∞, if for any sequence λi→∞ and normalized eigenfunctions ψi∈Vkλi, there exists a normalized eigenfunction ψ∈Vk such that ψi→ψ strongly in Hs(ℝN) along a subsequence. We have the following lemma.
Lemma 2.5.
We have μkλ→μk and Vkλ→Vk as λ→∞.
Proof.
We prove this lemma by induction.
Step 1. We firstly prove the case for k=1, i.e. up to a subsequence, we will prove that
λ
n
→
+
∞
,
μ
1
λ
n
→
μ
1
,
V
1
λ
n
→
V
1
as
n
→
+
∞
.
According to the definition of μ1λn, it is easy to see that μ1λn≤μ1. Now we assume that ψn∈Eλnis such that
(2.19)
∫
ℝ
N
ψ
n
2
𝑑
x
=
1
,
∫
ℝ
N
∫
ℝ
N
|
ψ
n
(
x
)
-
ψ
n
(
y
)
|
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
+
∫
ℝ
N
(
λ
n
a
(
x
)
-
δ
)
ψ
n
2
𝑑
x
=
μ
1
λ
n
.
By (2.19), we have
∥
ψ
n
∥
λ
n
2
=
∫
ℝ
N
∫
ℝ
N
|
ψ
n
(
x
)
-
ψ
n
(
y
)
|
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
+
∫
ℝ
N
V
λ
n
ψ
N
2
𝑑
x
+
∫
ℝ
N
V
λ
n
-
ψ
n
2
𝑑
x
=
μ
1
λ
n
+
∫
ℝ
N
V
λ
n
-
ψ
n
2
𝑑
x
≤
μ
1
+
δ
.
Then by Lemma 2.1, {ψn} is bounded in Hs(ℝN). Up to a subsequence, there is ψ∈Hs(ℝN) such that
(2.20)
{
ψ
n
⇀
ψ
weakly in
H
s
(
ℝ
N
)
,
ψ
n
→
ψ
strongly in
L
loc
2
(
ℝ
N
)
,
ψ
n
→
ψ
a.e. in
ℝ
N
as λn→+∞.
Claim 1.
We have ψ(x)=0 a.e. in RN∖Ω.
In fact, let Cm:={x∈ℝN:a(x)>1m}. For fixed positive integer m, by (2.19), we have
∫
C
m
ψ
2
𝑑
x
≤
m
λ
n
∫
ℝ
N
λ
n
a
(
x
)
ψ
n
2
𝑑
x
≤
m
λ
n
(
∫
ℝ
N
∫
ℝ
N
|
ψ
n
(
x
)
-
ψ
n
(
y
)
|
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
+
∫
ℝ
N
λ
n
a
(
x
)
ψ
n
2
𝑑
x
)
≤
m
λ
n
(
μ
1
+
δ
)
→
0
,
as λn→+∞. Thus u(x)=0 a.e. in Cm for m=1,2,3,…. Since ⋃m=1∞Cm=ℝN∖Ω, it follows that u(x)=0 a.e. in ℝN∖Ω.
Claim 2.
We have ∫Ωψ2𝑑x=1.
As in Section 1 (see (1.2) in Remark 1.1), Ω⊂BR(0) and a(x)>a0 for all |x|>R. Then by (2.19), we have
∫
ℝ
N
∖
B
R
(
0
)
ψ
n
2
𝑑
x
≤
1
a
0
λ
n
∫
ℝ
N
∖
B
R
(
0
)
λ
n
a
(
x
)
ψ
n
2
𝑑
x
≤
1
a
0
λ
n
(
∫
ℝ
N
∫
ℝ
N
|
ψ
n
(
x
)
-
ψ
n
(
y
)
|
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
+
∫
ℝ
N
λ
n
a
(
x
)
ψ
n
2
𝑑
x
)
=
1
a
0
λ
n
∫
ℝ
N
∫
ℝ
N
|
ψ
n
(
x
)
-
ψ
n
(
y
)
|
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
+
1
a
0
λ
n
∫
ℝ
N
λ
n
V
λ
n
ψ
n
2
𝑑
x
+
δ
a
0
λ
n
∫
ℝ
N
|
ψ
n
|
2
𝑑
x
≤
1
a
0
λ
n
(
μ
1
+
δ
)
→
0
as λn→+∞. Thus
(2.21)
lim
n
→
+
∞
∫
ℝ
N
∖
B
R
(
0
)
ψ
n
2
𝑑
x
=
0
.
Combine (2.19), (2.20), (2.21) and Claim 1 to have
∫
Ω
ψ
2
𝑑
x
=
∫
B
R
(
0
)
ψ
2
𝑑
x
=
lim
n
→
+
∞
∫
B
R
(
0
)
ψ
n
2
𝑑
x
=
lim
n
→
+
∞
∫
ℝ
N
ψ
n
2
𝑑
x
-
lim
n
→
+
∞
∫
ℝ
N
∖
B
R
(
0
)
ψ
n
2
𝑑
x
=
1
.
According to Claim 1 and Claim 2, we find that
(2.22)
∫
ℝ
N
ψ
2
𝑑
x
=
∫
Ω
ψ
2
𝑑
x
=
1
.
Let ψn1=ψn-ψ. By (2.20) and Brézis–Lieb’s lemma,
(2.23)
∫
ℝ
N
ψ
n
2
𝑑
x
=
∫
ℝ
N
ψ
2
𝑑
x
+
∫
ℝ
N
(
ψ
n
1
)
2
𝑑
x
+
o
(
1
)
and
(2.24)
∫
ℝ
N
∫
ℝ
N
|
ψ
n
(
x
)
-
ψ
n
(
y
)
|
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
=
∫
ℝ
N
∫
ℝ
N
|
ψ
(
x
)
-
ψ
(
y
)
|
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
+
∫
ℝ
N
∫
ℝ
N
|
ψ
n
1
(
x
)
-
ψ
n
1
(
y
)
|
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
+
o
(
1
)
as n→+∞. Thus by (2.19), (2.22) and (2.23), we have ψn→ψ strongly in L2(ℝN) as n→+∞. By Claim 1, we obtain that ψ∈E0. Thus by (2.15), (2.19), (2.22) and (2.24), we have
μ
1
:=
inf
u
∈
E
0
,
∥
u
∥
L
2
(
Ω
)
=
1
(
∫
ℝ
N
∫
ℝ
N
|
u
(
x
)
-
u
(
y
)
|
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
-
∫
Ω
δ
u
2
𝑑
x
)
≤
∫
ℝ
N
∫
ℝ
N
|
ψ
(
x
)
-
ψ
(
y
)
|
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
-
∫
Ω
δ
ψ
2
𝑑
x
≤
lim
n
→
∞
(
∫
ℝ
N
∫
ℝ
N
|
ψ
n
(
x
)
-
ψ
n
(
y
)
|
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
-
∫
ℝ
N
δ
ψ
n
2
𝑑
x
)
≤
lim
n
→
∞
μ
1
λ
n
≤
μ
1
,
which implies that μ1λn→μ1 and ψn→ψ in Hs(ℝN) as n→∞.
Step 2. Suppose k≥2 and the results hold up to k-1. We need to prove the same results hold for the k-th eigenvalues.
Since Lλψ=L0ψ for all ψ∈E0, by the k-th Rayleigh quotient descriptions of μkλ and μk, we have
lim sup
λ
→
+
∞
μ
k
λ
≤
μ
k
.
Just like in the case k=1, we can select λn→+∞ and the normalized eigenfunctions ψn∈Vkλn which is the eigenfunction space corresponding to μkλn satisfying
∫
ℝ
N
∫
ℝ
N
|
ψ
n
(
x
)
-
ψ
n
(
y
)
|
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
+
∫
ℝ
N
λ
n
(
a
(
x
)
-
δ
)
ψ
n
2
𝑑
x
=
μ
k
λ
n
and
∫
ℝ
N
ψ
n
2
𝑑
x
=
1
,
⊥
ψ
n
V
j
λ
n
,
j
=
1
,
2
,
3
,
…
,
k
-
1
.
Similar to the proof in Step 1, we have
{
ψ
n
⇀
ψ
weakly in
H
s
(
ℝ
N
)
,
ψ
n
→
ψ
strongly in
L
loc
2
(
ℝ
N
)
,
ψ
n
→
ψ
a.e. in
ℝ
N
for some ψ∈E0 satisfying ∫Ω|ψ|2𝑑x=1. Since ⊥ψnVjλn, j=1,2,…,k-1, and Vjλn→Vj as n→+∞, it follows that ⊥ψVj, j=1,2,…,k-1, and
μ
k
≤
∫
ℝ
N
∫
ℝ
N
|
ψ
(
x
)
-
ψ
(
y
)
|
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
-
∫
Ω
δ
ψ
2
𝑑
x
≤
lim
n
→
∞
(
∫
ℝ
N
∫
ℝ
N
|
ψ
n
(
x
)
-
ψ
n
(
y
)
|
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
-
∫
Ω
δ
ψ
n
2
𝑑
x
)
≤
lim
n
→
∞
μ
k
λ
n
≤
μ
k
.
This induces that μkλn→μk,Vkλn→Vk as n→+∞. ∎
Corollary 2.6.
For λ large, the operator (-Δ)s+λa+a0 on Eλ is nondegenerate and has finite Morse index dj:=dimEλ- uniformly in λ.
2.4 Modified Nehari–Pankov Manifold
Recall the modified Nehari–Pankov manifold
𝒩
λ
:=
{
u
∈
E
λ
∖
{
0
}
:
P
λ
-
∇
I
λ
(
u
)
=
0
,
I
λ
′
(
u
)
u
=
0
}
⊂
E
λ
∖
E
0
-
and the Nehari–Pankov manifold
𝒩
0
:=
{
u
∈
E
0
∖
{
0
}
:
P
0
-
∇
I
0
(
u
)
=
0
,
I
0
′
(
u
)
u
=
0
}
⊂
E
0
∖
E
0
-
.
The corresponding levels are
c
λ
:=
inf
u
∈
𝒩
λ
I
λ
(
u
)
and
c
0
:=
inf
u
∈
𝒩
0
I
0
(
u
)
.
For the Nehari–Pankov manifold 𝒩0, we have the following lemma due to Szulkin and Weth [29].
Lemma 2.7.
For any ω∈E0∖E0-, set
H
ω
:=
{
v
+
t
ω
:
v
∈
E
0
-
,
t
>
0
}
.
The following properties hold:
𝒩
0
=
{
w
∈
E
0
∖
E
0
-
:
∇
(
I
0
|
H
w
)
=
0
}
.
For every
w
∈
E
0
+
∖
{
0
}
there exist
t
w
>
0
and
φ
(
w
)
∈
E
0
-
such that
H
w
∩
𝒩
0
=
{
φ
(
w
)
+
t
w
⋅
w
}
.
For every
w
∈
𝒩
0
and every
u
∈
H
w
∖
{
w
}
there holds
I
0
(
u
)
<
I
0
(
w
)
.
c
0
=
inf
u
∈
𝒩
0
I
0
(
u
)
>
0
.
Proof.
The proof of this lemma is similar to the proofs of [29, Lemma 2.2–2.6]; we omit the details. ∎
For the modified Nehari–Pankov manifold, we have the following lemma.
Lemma 2.8.
For any ω∈Eλ∖E0-, set
H
^
ω
:=
{
v
+
t
ω
:
v
∈
E
0
-
,
t
>
0
}
.
The following properties hold:
𝒩
λ
=
{
w
∈
E
λ
∖
E
0
-
:
∇
(
I
λ
|
H
^
w
)
=
0
}
.
Let
E
^
λ
+
:=
{
ω
∈
E
λ
+
:
∫
ℝ
N
ω
e
i
𝑑
x
=
0
,
i
=
1
,
2
,
…
,
k
}
.
Then for any
w
∈
E
^
λ
+
∖
{
0
}
there exist
t
w
>
0
and
φ
(
w
)
∈
E
0
-
such that
H
^
w
∩
𝒩
λ
=
{
φ
(
w
)
+
t
w
⋅
w
}
.
For any
w
∈
𝒩
λ
and
u
∈
H
^
w
∖
{
w
}
there holds
I
λ
(
u
)
<
I
λ
(
w
)
.
c
λ
=
inf
u
∈
𝒩
λ
I
λ
(
u
)
≥
τ
>
0
for some small
τ
>
0
independent of large λ.
Proof.
The proofs of (i)–(iii) are obvious. We just need to prove (iv).
Firstly, we claim that there exists a μ0>0 such that for any λ>μ0 and u∈Eλ+, we have
∫
ℝ
2
N
(
u
(
x
)
-
u
(
y
)
)
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
+
∫
ℝ
N
V
λ
u
2
𝑑
x
≥
C
(
∫
ℝ
2
N
(
u
(
x
)
-
u
(
y
)
)
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
+
∫
ℝ
N
V
λ
+
u
2
𝑑
x
)
for some C>0 which is independent of λ.
In fact, for any u∈Eλ+, we have u=uλ++uλ-, uλ+∈Eλ+, uλ-∈Eλ-. Note that
∫
ℝ
2
N
(
u
λ
+
(
x
)
-
u
λ
+
(
y
)
)
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
+
∫
ℝ
N
V
λ
+
(
u
λ
+
)
2
𝑑
x
=
∫
ℝ
2
N
(
u
λ
+
(
x
)
-
u
λ
+
(
y
)
)
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
+
∫
ℝ
N
V
λ
(
u
λ
+
)
2
𝑑
x
+
∫
ℝ
N
V
λ
-
(
u
λ
+
)
2
𝑑
x
≤
μ
k
+
1
(
L
λ
)
+
δ
μ
k
+
1
(
L
λ
)
∫
ℝ
2
N
(
u
λ
+
(
x
)
-
u
λ
+
(
y
)
)
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
+
∫
ℝ
N
V
λ
(
u
λ
+
)
2
𝑑
x
.
Then a simple computation shows
∫
ℝ
2
N
(
u
(
x
)
-
u
(
y
)
)
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
+
∫
ℝ
N
V
λ
u
2
𝑑
x
=
∫
ℝ
2
N
(
u
λ
+
(
x
)
-
u
λ
+
(
y
)
)
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
+
∫
ℝ
N
V
λ
(
u
λ
+
)
2
𝑑
x
+
∫
ℝ
2
N
(
u
λ
-
(
x
)
-
u
λ
-
(
y
)
)
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
+
∫
ℝ
N
V
λ
(
u
λ
-
)
2
𝑑
x
≥
μ
k
+
1
(
L
λ
)
μ
k
+
1
(
L
λ
)
+
δ
(
∫
ℝ
2
N
(
u
λ
+
(
x
)
-
u
λ
+
(
y
)
)
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
+
∫
ℝ
N
V
λ
+
(
u
λ
+
)
2
𝑑
x
)
+
∫
ℝ
2
N
(
u
λ
-
(
x
)
-
u
λ
-
(
y
)
)
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
+
∫
ℝ
N
V
λ
(
u
λ
-
)
2
𝑑
x
≥
μ
k
+
1
(
L
λ
)
μ
k
+
1
(
L
λ
)
+
δ
(
∫
ℝ
2
N
(
u
(
x
)
-
u
(
y
)
)
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
+
∫
ℝ
N
V
λ
+
u
2
𝑑
x
)
+
{
δ
μ
k
+
1
(
L
λ
)
+
δ
(
∫
ℝ
2
N
(
u
λ
-
(
x
)
-
u
λ
-
(
y
)
)
2
|
x
-
y
|
N
+
2
s
d
x
d
y
+
∫
ℝ
N
V
λ
+
(
u
λ
-
)
2
d
x
)
-
μ
k
+
1
(
L
λ
)
μ
k
+
1
(
L
λ
)
+
δ
∫
ℝ
N
V
λ
-
[
(
u
λ
-
)
2
+
2
u
λ
+
u
λ
-
]
d
x
}
,
since
u
λ
-
=
∑
i
=
1
k
(
∫
ℝ
N
u
e
λ
,
i
𝑑
x
)
e
λ
,
i
=
∑
i
=
1
k
[
∫
ℝ
N
u
(
e
λ
,
i
-
e
i
)
𝑑
x
]
e
λ
,
i
,
where ei and eλ,i are the eigenfunction of L0 and Lλ, respectively. Note that eλ,i→ei in Hs(ℝN) for each i as λ→+∞. Then we can easily get
∫
ℝ
2
N
(
u
λ
-
(
x
)
-
u
λ
-
(
y
)
)
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
+
∫
ℝ
N
V
λ
(
u
λ
-
)
2
𝑑
x
+
∫
ℝ
N
V
λ
-
[
(
u
λ
-
)
2
+
2
u
λ
+
u
λ
-
]
𝑑
x
=
o
(
1
)
∥
u
∥
λ
2
.
Since μk+1(Lλ)→μk+1(L0) as λ→+∞, there exists μ0>λ0 such that for any λ>μ0, we have
∫
ℝ
2
N
(
u
(
x
)
-
u
(
y
)
)
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
+
∫
ℝ
N
V
λ
u
2
𝑑
x
≥
μ
k
+
1
(
L
0
)
μ
k
+
1
(
L
0
)
+
δ
(
∫
ℝ
2
N
(
u
(
x
)
-
u
(
y
)
)
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
+
∫
ℝ
N
V
λ
+
u
2
𝑑
x
)
.
Secondly, let
S
α
:=
{
u
∈
E
λ
+
:
∫
ℝ
2
N
(
u
(
x
)
-
u
(
y
)
)
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
+
∫
ℝ
N
V
λ
u
2
𝑑
x
=
α
2
}
.
For any u∈Sα, by the Sobolev inequality, we have
∫
ℝ
N
|
u
|
p
≤
C
[
∫
ℝ
2
N
(
u
(
x
)
-
u
(
y
)
)
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
+
∫
ℝ
N
V
λ
+
u
2
𝑑
x
]
p
2
≤
C
[
∫
ℝ
2
N
(
u
(
x
)
-
u
(
y
)
)
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
+
∫
ℝ
N
V
λ
u
2
𝑑
x
]
p
2
≤
C
α
p
.
Therefore, for α>0 small enough, we have Iλ(u)≥12α2-Cαp≥14α2>0. This implies that infSαIλ(u)>0.
Finally, for any ω∈𝒩λ, ω+=ω-ω-∈H^ω, take t>0 small enough such that tω+∈Hω∩Sα, thus take τ-14α2. By (iii), we have
I
λ
(
ω
)
>
I
λ
(
t
ω
+
)
≥
inf
S
α
I
λ
(
u
)
≥
τ
>
0
,
which implies that cλ≥τ>0. ∎
At the end of this subsection, in the following lemma we will prove that the minimizer for cλ is indeed a least energy solution for (${\mathcal{P}_{\lambda}}$).
Lemma 2.9.
For λ large enough, u∈Nλ is an achieved function for cλ, i.e. cλ=Iλ(u). Then u is a least energy solution of (${\mathcal{P}_{\lambda}}$).
Proof.
Note that each solution for (${\mathcal{P}_{\lambda}}$) belongs to Nλ; thus we just need to show that the achieved function u is indeed a solution.
We denote
G
0
(
u
)
=
I
λ
′
(
u
)
u
,
G
i
(
u
)
=
I
λ
′
(
u
)
e
i
,
i
=
1
,
2
,
…
,
k
.
Then the modified Nehari–Pankov manifold
𝒩
λ
=
{
u
∈
E
λ
∖
{
0
}
:
G
i
(
u
)
=
0
,
i
=
1
,
2
,
…
,
k
}
.
According to the Lagrange Multiplier Theorem, there exists (λ0,λ1,…,λk)∈ℝk such that
I
λ
′
(
u
)
+
λ
0
G
0
′
(
u
)
+
λ
1
G
1
′
(
u
)
+
⋯
+
λ
k
G
k
′
(
u
)
=
0
.
Multiplying u and ei on both sides of the above equation, respectively, for i=1,2,…,k, we have the following system:
{
a
00
λ
0
+
a
0
,
1
λ
1
+
⋯
+
a
0
k
λ
k
=
0
,
a
10
λ
0
+
a
1
,
1
λ
1
+
⋯
+
a
1
k
λ
k
=
0
,
⋮
a
k
0
λ
0
+
a
k
,
1
λ
1
+
⋯
+
a
k
k
λ
k
=
0
,
where
a
00
=
(
p
-
2
)
∫
ℝ
N
|
u
|
p
𝑑
x
,
a
i
i
=
(
p
-
1
)
∫
ℝ
N
|
u
|
p
-
2
u
e
i
2
𝑑
x
-
μ
i
(
L
0
)
∫
ℝ
N
e
i
2
𝑑
x
,
a
0
,
i
=
a
i
,
0
=
(
p
-
2
)
∫
ℝ
N
|
u
|
p
-
2
u
e
i
𝑑
x
,
a
i
j
=
a
j
i
=
(
p
-
1
)
∫
ℝ
N
|
u
|
p
-
2
e
i
e
j
𝑑
x
for i,j=1,2,…,k. It is easy to show that the coefficient matrix of the above system is positive definite. Thus the solution of the above system is (λ0,λ1,…,λk)=(0,0,…,0) which implies that u is indeed a solution to (${\mathcal{P}_{\lambda}}$). ∎
3 Existence of Least Energy Solutions to (1.3)
In this section, we present an existence result for the least energy solutions to (1.3). Recall the energy functional of (1.3):
I
0
(
u
)
=
1
2
∫
ℝ
N
∫
ℝ
N
|
u
(
x
)
-
u
(
y
)
|
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
-
1
2
∫
Ω
δ
u
2
𝑑
x
-
1
p
∫
Ω
|
u
|
p
𝑑
x
for all
u
∈
E
0
.
And the Nehari–Pankov manifold related to I0 is
𝒩
0
=
{
u
∈
E
0
∖
{
0
}
:
P
0
-
∇
I
0
(
u
)
=
0
,
I
0
′
(
u
)
u
=
0
}
,
with the corresponding least level
c
0
:=
inf
𝒩
0
I
0
(
u
)
.
We state our main result in this section as follows.
Proposition 3.1.
Assume that Ω:=int{x∈RN:a(x)=0} is a smooth bounded domain in RN, c0:=infN0I0(u) and one of the following conditions hold:
N
>
2
s
, p is subcritical, i.e. 2<p<2s*,
N
≥
4
s
, p is critical, i.e. p=2s*.
Then c0 is achieved by a sign-changing function which is a least energy solution to (1.3).
To show that, we get a (PS)c0 sequence by Ekeland’s variational principle and study the boundedness and compactness of the (PS)c0 sequence. We have the following lemmas.
Lemma 3.2.
Let 2<p≤2s*, N>2s. Then there exists a (PS)c0 sequence {un}n≥1, i.e.
I
0
(
u
n
)
→
c
0
,
I
0
′
(
u
n
)
→
0
as
n
→
+
∞
.
Proof.
Indeed, similar arguments can be fund in the proof of [22, Theorem 5.7]. This is a direct result of Ekeland’s variational principle. For the convenience of the reader, here we give a sketch of the argument.
By the definition of c0, we can easily get a minimizing sequence {vn}⊂𝒩0 satisfying
I
0
(
v
n
)
→
c
0
as
n
→
∞
.
Take
ϵ
n
=
{
I
0
(
v
n
)
-
c
0
,
if
I
0
(
v
n
)
>
c
0
,
1
n
,
if
I
0
(
v
n
)
=
c
0
.
Then by Ekeland’s variational principle, there exists a sequence {un}⊂E0 such that
I
0
(
u
n
)
≤
I
0
(
v
n
)
,
∥
u
n
-
v
n
∥
0
≤
ϵ
n
1
2
,
∥
(
I
0
′
(
u
n
)
)
1
∥
E
0
′
≤
ϵ
n
1
2
,
where E0′ is the dual space of E0 and (I0′(vn))1 denotes the orthogonal projection of I0′(un) onto the tangent space of 𝒩0 at un. One can show that there is a constant C>0 which is independent of n such that
∥
I
0
′
(
u
n
)
∥
E
0
′
≤
C
∥
(
I
0
′
(
u
n
)
)
1
∥
E
0
′
.
Let n→+∞; we have I0(un)→c0, I0′(un)→0 as n→+∞. ∎
Lemma 3.3.
Let 2<p≤2s*, N>2s, {un}n≥1 be a (PS)c0 sequence. Then {un}n≥1 is bounded in E0.
Proof.
According to the definition of (PS)c0 sequences, there exists a positive integer number M such that for each n>M, we have
(3.1)
c
0
+
1
+
∥
u
n
∥
0
≥
I
0
(
u
n
)
-
1
2
I
0
′
(
u
n
)
u
n
=
(
1
2
-
1
p
)
∫
Ω
|
u
n
|
p
𝑑
x
and
(3.2)
c
0
+
1
+
∥
u
n
∥
0
≥
I
0
(
u
n
)
-
1
p
I
0
′
(
u
n
)
u
n
=
p
-
2
2
p
(
∫
ℝ
N
∫
ℝ
N
(
u
(
x
)
-
u
(
y
)
)
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
-
δ
∫
Ω
u
2
𝑑
x
)
.
By Hölder’s inequality, we have
(3.3)
∫
Ω
u
2
𝑑
x
≤
|
Ω
|
1
-
2
p
(
∫
Ω
|
u
|
p
𝑑
x
)
2
p
.
Combining (3.1)–(3.3), we can easily get that for each n>M,
∥
u
n
∥
0
2
=
∫
ℝ
N
∫
ℝ
N
(
u
(
x
)
-
u
(
y
)
)
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
≤
C
(
c
0
+
1
+
∥
u
n
∥
0
)
,
where the constant C does not depend on n. Thus {un}n≥1 is bounded in E0. ∎
Lemma 3.4.
Assume that one of the following conditions holds:
N
>
2
s
, p is subcritical, i.e. 2<p<2s*,
N
≥
4
s
, p is critical, i.e. p=2s*.
Then the (PS)c0 condition holds. Namely, let {un}n≥1 be a (PS)c0 sequence. Then there must exist u∈E0 such that, up to a subsequence, un→u strongly in E0 as n→+∞.
Proof.
We consider two cases.
Case 1: p is subcritical, i.e. 2<p<2s*. In fact, by Lemma 3.3, there exists a function u∈E0, up to a subsequence, still denoted by {un}n≥1, such that
(3.4)
{
u
n
⇀
u
weakly in
E
0
,
u
n
→
u
strongly in
L
2
(
Ω
)
and
L
p
(
Ω
)
,
u
n
→
u
a.e. in
Ω
as n→+∞. According to the definition of (PS)c0 sequence and (3.4), we have
(3.5)
o
(
1
)
=
(
I
0
′
(
u
n
)
-
I
0
′
(
u
)
)
(
u
n
-
u
)
(3.6)
=
∫
ℝ
N
∫
ℝ
N
(
(
u
n
-
u
)
(
x
)
-
(
u
n
-
u
)
(
y
)
)
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
-
δ
∫
Ω
(
u
n
-
u
)
2
𝑑
x
-
∫
Ω
(
|
u
n
|
p
-
2
u
n
-
|
u
|
p
-
2
u
)
(
u
n
-
u
)
𝑑
x
.
By (3.4) and the Lebesgue Dominated Theorem, we can easily get that
|
u
n
|
p
-
2
u
n
→
|
u
|
p
-
2
u
strongly in
L
p
p
-
1
(
Ω
)
.
By Hölder’s inequality and (3.4), we have
(3.7)
|
∫
Ω
(
|
u
n
|
p
-
2
u
n
-
|
u
|
p
-
2
u
)
(
u
n
-
u
)
𝑑
x
|
≤
(
∫
Ω
|
|
u
n
|
p
-
2
u
n
-
|
u
|
p
-
2
u
|
p
p
-
1
𝑑
x
)
p
-
1
p
(
∫
Ω
|
u
n
-
u
|
p
𝑑
x
)
1
p
→
0
as n→+∞. Combining (3.4), (3.6) and (3.7), we have as n→+∞,
∫
ℝ
N
∫
ℝ
N
(
(
u
n
-
u
)
(
x
)
-
(
u
n
-
u
)
(
y
)
)
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
→
0
.
Thus limn→+∞∥un-u∥0=0.
Case 2: p is critical, i.e. p=2s*. To begin with, by the next lemma, we see
(3.8)
0
<
c
0
<
s
N
S
N
2
s
.
As proved in Lemma 3.3, {un} in bounded in E0. Taking if necessary a subsequence, we may assume that
(3.9)
{
u
n
⇀
u
in
E
0
and
L
2
s
*
(
Ω
)
,
u
n
→
u
strongly in
L
2
(
Ω
)
,
u
n
→
u
a.e. in
Ω
.
By (3.9) and the definition of (PS)c0 sequences, for each v∈E0, we have
(3.10)
0
=
lim
n
→
+
∞
I
0
′
(
u
n
)
v
(3.11)
=
lim
n
→
+
∞
∫
ℝ
N
∫
ℝ
N
(
u
n
(
x
)
-
u
n
(
y
)
)
(
v
(
x
)
-
v
(
y
)
)
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
-
δ
lim
n
→
+
∞
∫
Ω
u
n
v
𝑑
x
-
lim
n
→
+
∞
∫
Ω
|
u
n
|
2
s
*
-
2
u
n
v
𝑑
x
(3.12)
=
∫
ℝ
N
∫
ℝ
N
(
u
(
x
)
-
u
(
y
)
)
(
v
(
x
)
-
v
(
y
)
)
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
-
δ
∫
Ω
u
v
𝑑
x
-
∫
Ω
|
u
|
2
s
*
-
2
u
v
𝑑
x
,
i.e. I0′(u)=0. By (3.12), we have
(3.13)
I
0
(
u
)
=
I
0
(
u
)
-
1
2
I
0
′
(
u
)
u
=
(
1
2
-
1
2
s
*
)
∫
Ω
|
u
|
2
s
*
𝑑
x
≥
0
.
Let vn=un-u, due to Brézis–Lieb’s lemma and (3.9), we have as n→+∞,
(3.14)
∥
u
n
∥
0
2
=
∥
v
n
∥
0
2
+
∥
u
∥
0
2
+
o
(
1
)
,
∥
u
n
∥
L
2
s
*
(
Ω
)
2
s
*
=
∥
v
n
∥
L
2
s
*
(
Ω
)
2
s
*
+
∥
u
∥
L
2
s
*
(
Ω
)
2
s
*
.
By (3.9) and (3.14), we have
(3.15)
o
(
1
)
=
〈
I
0
′
(
u
n
)
,
u
n
〉
(3.16)
=
∫
ℝ
N
∫
ℝ
N
(
u
n
(
x
)
-
u
n
(
y
)
)
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
-
δ
∫
Ω
u
n
2
𝑑
x
-
∫
Ω
|
u
n
|
2
s
*
𝑑
x
(3.17)
=
∫
ℝ
N
∫
ℝ
N
(
u
(
x
)
-
u
(
y
)
)
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
-
δ
∫
Ω
u
2
𝑑
x
-
∫
Ω
|
u
|
2
s
*
𝑑
x
+
∫
ℝ
N
∫
ℝ
N
(
v
n
(
x
)
-
v
n
(
y
)
)
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
-
∫
Ω
|
v
n
|
2
s
*
𝑑
x
+
o
(
1
)
(3.18)
=
∫
ℝ
N
∫
ℝ
N
(
v
n
(
x
)
-
v
n
(
y
)
)
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
-
∫
Ω
|
v
n
|
2
s
*
𝑑
x
+
o
(
1
)
and
(3.19)
I
0
(
u
n
)
=
1
2
∫
ℝ
N
∫
ℝ
N
(
u
n
(
x
)
-
u
n
(
y
)
)
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
-
δ
2
∫
Ω
u
n
2
𝑑
x
-
1
2
s
*
∫
Ω
|
u
n
|
2
s
*
𝑑
x
=
(
1
2
∫
ℝ
N
∫
ℝ
N
(
u
(
x
)
-
u
(
y
)
)
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
-
δ
2
∫
Ω
u
2
𝑑
x
-
1
2
s
*
∫
Ω
|
u
|
2
s
*
𝑑
x
)
(3.20)
+
1
2
∫
ℝ
N
∫
ℝ
N
(
v
n
(
x
)
-
v
n
(
y
)
)
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
-
1
2
s
*
∫
Ω
|
v
n
|
2
s
*
𝑑
x
+
o
(
1
)
(3.21)
=
I
0
(
u
)
+
1
2
∫
ℝ
N
∫
ℝ
N
(
v
n
(
x
)
-
v
n
(
y
)
)
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
-
1
2
s
*
∫
Ω
|
v
n
|
2
s
*
𝑑
x
+
o
(
1
)
.
We assume that b=limn→+∞∫Ω|vn|2s*𝑑x.
(i) If b=0, we complete the proof by (3.18).
(ii) If b>0, by (3.18) and Sobolev inequality, we have
b
=
lim
n
→
+
∞
∫
Ω
|
v
n
|
2
s
*
𝑑
x
=
lim
n
→
+
∞
∫
ℝ
N
∫
ℝ
N
(
v
n
(
x
)
-
v
n
(
y
)
)
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
≥
S
(
lim
n
→
+
∞
∫
Ω
|
v
n
|
2
s
*
𝑑
x
)
2
2
s
*
=
S
b
2
2
s
*
.
Thus b≥SN2s. On the other hand, by (3.13), (3.18) and (3.21), we have
c
0
=
lim
n
→
+
∞
I
0
(
u
n
)
≥
1
2
lim
n
→
+
∞
∫
ℝ
N
∫
ℝ
N
(
v
n
(
x
)
-
v
n
(
y
)
)
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
-
1
2
s
*
lim
n
→
+
∞
∫
Ω
|
v
n
|
2
s
*
𝑑
x
=
(
1
2
-
1
2
s
*
)
b
.
Thus b<SN2s due to the estimate of c0, see (3.8). This leads to a contradiction. ∎
We prove (3.8) by the following lemma.
Lemma 3.5.
Assume that N≥4s, p=2s* is critical and S is defined as (2.5) in Section 2.2. Then
0
<
c
0
<
s
N
S
N
2
s
.
Proof.
According to (iv) in Lemma 2.7, we know that c0>0. We just need to verify that c0<sNSN2s. Without loss of generality, we may assume that 0∈Ω and B4α⊆Ω for some α>0. Let
u
^
=
U
(
x
)
∥
U
(
x
)
∥
L
2
s
*
(
ℝ
N
)
,
where
U
(
x
)
:=
κ
(
μ
2
+
|
x
|
2
)
-
N
-
2
s
2
.
As we have described in Section 2.2, u^ is a minimizer for S. Let
U
ϵ
(
x
)
=
ϵ
2
s
-
N
2
u
*
(
x
ϵ
)
,
where u*(x)=u^(xS12s) and ϵ>0 is small. Put uϵ=η(x)Uϵ(x), where η(x) is a smooth cutoff function satisfying
η
=
1
in
B
α
,
η
=
0
in
ℝ
N
∖
B
2
α
,
0
≤
η
≤
1
.
By a standard argument which is similar to [25, Proposition 7.2], we have for N≥4s and ϵ>0 small enough,
(3.22)
∫
ℝ
N
∫
ℝ
N
(
u
ϵ
(
x
)
-
u
ϵ
(
y
)
)
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
-
δ
∫
Ω
|
u
ϵ
(
x
)
|
2
𝑑
x
(
∫
Ω
|
u
ϵ
|
2
s
*
𝑑
x
)
2
2
s
*
<
S
.
Let Huϵ:=span{E0-,uϵ}. According to [25, Proposition 7.3] and some calculation, we have
M
ϵ
:=
max
{
∫
ℝ
N
∫
ℝ
N
(
u
(
x
)
-
u
(
y
)
)
2
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
-
δ
∫
Ω
|
u
(
x
)
|
2
𝑑
x
:
u
∈
H
u
ϵ
∖
{
0
}
,
∫
Ω
|
u
ϵ
|
2
s
*
𝑑
x
=
1
}
<
S
for ϵ>0 small and N≥4s. For each u∈Huϵ∖{0}, we have
I
0
(
u
)
≤
max
t
≥
0
I
0
(
t
u
)
=
s
N
(
∥
u
∥
0
2
-
δ
∥
u
∥
L
2
(
Ω
)
2
∥
u
∥
L
2
s
*
(
Ω
)
2
)
.
Thus
max
u
∈
H
u
ϵ
I
0
(
u
)
≤
s
N
M
ϵ
N
2
<
s
N
S
N
2
s
for ϵ>0 small and N≥4s. Lemma 2.7 immediately implies that c0<sNSN2s for N≥4s. ∎
Now we come to give the proof of Proposition 3.1.
Proof of Proposition 3.1.
According to the above lemmas, there is a sequence {un}⊂E0 such that I0(un)→c0, I0′(un)→0, and un→u strongly in E0. This implies that I0(u)=c0, I0′(u)=0. Thus u is a least energy solution to (1.3).
For δ≥λ1, u is sign-changing. In fact, let e1 be the principle eigenfunction corresponding to the principle eigenvalue λ1 of L0 defined in E0. We may assume that e1 is positive. Multiplying e1 on both sides of equation (1.3) and integrate over ℝN, we have
∫
ℝ
N
∫
ℝ
N
(
u
(
x
)
-
u
(
y
)
)
(
e
1
(
x
)
-
e
1
(
y
)
)
|
x
-
y
|
N
+
2
s
𝑑
x
𝑑
y
-
δ
∫
Ω
u
e
1
𝑑
x
=
∫
Ω
|
u
|
p
-
2
u
e
1
𝑑
x
.
This implies that
(3.23)
(
λ
1
-
δ
)
∫
Ω
u
e
1
𝑑
x
=
∫
Ω
|
u
|
p
-
2
u
e
1
𝑑
x
.
If u keeps sign, we may assume that u≥0. Thus the left side of (3.23) is nonpositive, but the right side is positive. This leads to a contradiction. Thus u changes sign. This completes the proof of Proposition 3.1. ∎
Remark 3.6.
Assume Ω=Ω1∪Ω2 is the interior part of the zero set a-1(0), where Ω1 and Ω2 are two smooth, connected and bounded domains in ℝN with Ω1∩Ω2≠∅. Suppose u is a least energy solution of (1.3) with u(x)=0 in Ω1 and u(x)≠0 in Ω2. Then for any x∈Ω1,
(
-
Δ
)
s
u
(
x
)
=
∫
Ω
2
-
u
(
y
)
|
x
-
y
|
N
+
2
s
𝑑
y
might be nonzero. For example, for 0<δ<λ1, u is nontrivial and keeps sign in Ω2. On one hand,
(
-
Δ
)
s
u
(
x
)
=
∫
Ω
2
-
u
(
y
)
|
x
-
y
|
N
+
2
s
𝑑
y
is positive or negative. But on the other hand, for x∈Ω1,
(
-
Δ
)
s
u
(
x
)
=
δ
u
(
x
)
+
u
(
x
)
p
-
1
=
0
.
Thus u(x)≠0 in both Ω1 and Ω2. However, for the Laplacian, the local case, u=0 in Ω if and only if Δu=0. This is the difference between the local and nonlocal problems.
Remark 3.7.
Recently, Raffaella Servadei and Enrico Valdinoci considered the following problem:
(3.24)
{
(
-
Δ
)
s
u
(
x
)
-
δ
u
(
x
)
=
f
(
x
,
u
(
x
)
)
in
Ω
,
u
(
x
)
=
0
in
ℝ
N
∖
Ω
,
where 0<s<1, N>2s, and Ω⊆ℝN is a bounded domain with Lipschitz boundary and the nonlinearity f(x,u) satisfies some growth conditions. When f(x,u) is subcritical, if δ=0, they proved in [24] the existence of ground states to (3.24) by the Mountain Pass Theorem. If δ≠λ1, they obtained in [26] a positive solution to (3.24) by the Mountain Pass Theorem for δ<λ1 and a sign-changing solution by the Linking Theorem for δ≥λ1. When f(x,u) is critical, the authors also obtained the similar results to (3.24). For more details, please see [27] and [25].
and
Lushun Wang
