1 Introduction
In this paper, we study the existence of a positive solution of the Choquard equation. More precisely, we consider the problem
(P)
{
-
Δ
u
=
(
∫
Ω
|
u
(
y
)
|
2
μ
*
|
x
-
y
|
μ
𝑑
y
)
|
u
|
2
μ
*
-
2
u
in
Ω
,
u
=
0
on
∂
Ω
,
where Ω is a smooth bounded domain in ℝN(N≥3), 2μ*=2N-μN-2, 0<μ<N.
The work on elliptic equations involving critical Sobolev exponent over non-contractible domains was initiated by J.-M. Coron in 1983. Indeed, Coron [10] proved the existence of a positive solution of the following critical elliptic problem
(Q)
{
-
Δ
u
=
u
N
+
2
N
-
2
in
Ω
,
u
>
0
in
Ω
,
u
=
0
on
∂
Ω
,
where Ω is a smooth bounded domain in ℝN and satisfies the following conditions: there exist constants 0<R1<R2<∞ such that
(1.1)
{
x
∈
ℝ
N
:
R
1
<
|
x
|
<
R
2
}
⊂
Ω
,
{
x
∈
ℝ
N
:
|
x
|
<
R
1
}
⊈
Ω
¯
.
Later on, Bahri and Coron [1] proved that if there exists a positive integer d such that Hd(Ω,ℤ2)≠0 (where Hd(Ω,ℤ2) the homology of dimension d of Ω with ℤ2 coefficients), then problem (Q) has a positive solution.
Benci and Cerami [4] considered the equation
(1.2)
{
-
Δ
u
+
λ
u
=
u
p
-
1
in
Ω
,
u
>
0
in
Ω
,
u
=
0
on
∂
Ω
,
where Ω⊂ℝN, N≥3, is a smooth bounded domain and 2<p<2*, λ∈ℝ+∪{0}. With the help of Ljusternik–Schnirelmann theory, Benci and Cerami showed that there exists a function λ¯:(2,2*)→ℝ+∪{0} such that for all λ≥λ¯(p), problem (1.2) has at least cat(Ω) distinct solutions. We cite [3, 5, 6, 11, 23, 27, 33, 37] and the references therein for the work on the existence of solutions over a non-contractible domain.
We recall that the Choquard equation (1.3) was first introduced in the pioneering work of Fröhlich [13] and Pekar [30] for the modeling of quantum polaron:
(1.3)
-
Δ
u
+
u
=
(
1
|
x
|
*
|
u
|
2
)
u
in
ℝ
3
.
As pointed out by Fröhlich [13] and Pekar, this model corresponds to the study of free electrons in an ionic lattice interact with phonons associated to deformations of the lattice or with the polarization that it creates on the medium (interaction of an electron with its own hole). In the approximation to Hartree–Fock theory of one component plasma, Choquard used equation (1.3) to describe an electron trapped in its own hole.
The Choquard equation is also known as the Schrödinger–Newton equation in models coupling the Schrödinger equation of quantum physics together with nonrelativistic Newtonian gravity. The equation can also be derived from the Einstein–Klein–Gordon and Einstein–Dirac system. Such a model was proposed for boson stars and for the collapse of galaxy fluctuations of scalar field dark matter. We refer for details to Elgart and Schlein [12], Giulini and Großardt [17], Jones [19], and Schunck and Mielke [34]. Penrose [31, 32] proposed equation (1.3) as a model of self-gravitating matter in which quantum state reduction was understood as a gravitational phenomenon.
As pointed out by Lieb [20], Choquard used equation (1.3) to study steady states of the one component plasma approximation in the Hartree–Fock theory. Classification of solutions of (1.3) was first studied by Ma and Zhao [22]. For a broad survey of Choquard equations we refer to Moroz and Van Schaftingen [26] and references therein. We also refer to Battaglia and Van Schaftingen [2], Cassani and Zhang [9], Mingqi, Rădulescu and Zhang [25], and Seok [35] as recent relevant contributions to the study of Choquard-type problems.
Recently, Gao and Yang [16] studied the Brezis–Nirenberg-type result for the following problem:
(1.4)
{
-
Δ
u
=
λ
u
+
(
∫
Ω
|
u
(
y
)
|
2
μ
*
|
x
-
y
|
μ
𝑑
y
)
|
u
|
2
μ
*
-
2
u
in
Ω
,
u
=
0
on
∂
Ω
,
where 0<λ, 0<μ<N, 2μ*=2N-μN-2, Ω is a smooth bounded domain in ℝN and 2μ* is the critical exponent in the sense of the Hardy–Littlewood–Sobolev inequality (2.1). They proved the Pohozaev identity for equation (1.4) and used variational methods and the minimizers of the best constant SH,L (defined in (2.3)) to show the existence, non-existence of solution depending on the range of λ. We cite [15, 14] for the Choquard equation with critical exponent in the sense of the Hardy–Littlewood–Sobolev inequality. However, the existence and multiplicity of solutions of nonlocal equations over non-contractible domains is still an open question. Therefore, it is essential to study the existence of a positive solution of elliptic equations involving convolution-type nonlinearity in non-contractible domains.
Inspiring by these results, we study in the present article the Coron problem for problem (P). More precisely, we show the existence of a high-energy positive solution in a non-contractible bounded domain particularly an annulus when the inner hole is sufficiently small. The functional associated with (P) is not C2 when μ>min{4,N} and this makes problem (P) more challenging.
In order to achieve the desired aim we first prove the non-existence result using the Pohozaev identity for Choquard equation on ℝ+N. We also prove the global compactness lemma for Choquard equation in bounded domains. In case of μ=0, such a lemma has been proved by Struwe [36] and later generalized to the p-Laplacian case by Mercuri and Willem [24]. In case of 0<μ<N, the method of defining Lévy concentration function is not useful. In the present article we gave the proof of global compactness Lemma 4.5 by introducing the notion of Morrey spaces. Finally, by using the concentration-compactness principle together with the deformation lemma, we prove the existence of high-energy positive solution. To the best of our knowledge, there is no work on Coron’s problem for Choquard equation.
We now state the main result of this paper.
Theorem 1.1.
Assume that Ω is a bounded domain in RN satisfying condition (1.1). If R2R1 is sufficiently large, then problem (P) admits a positive high-energy solution.
Turning to the layout of the paper, in Section2 we assemble notations and preliminary results. In Section3, we give the classification of all nonnegative solutions of Choquard equation. In Section4, we analyze the Palais–Smale sequences. In Section5, we prove our main result Theorem 1.1. We refer to the recent monograph by Papageorgiou, Rădulescu and Repovš [29] for some of the basic analytic tools used in this paper.
2 Preliminary Results
This section is devoted to the variational formulation, Pohozaev identity and non-existence result. The outset of the variational framework starts from the following Hardy–Littlewood–Sobolev inequality. We refer to Lieb and Loss [21] for more details.
Proposition 2.1.
Let t,r>1 and 0<μ<N with 1t+μN+1r=2, f∈Lt(RN) and h∈Lr(RN). There exists a sharp constant C(t,r,μ,N) independent of f,h such that
(2.1)
∫
ℝ
N
∫
ℝ
N
f
(
x
)
h
(
y
)
|
x
-
y
|
μ
𝑑
x
𝑑
y
≤
C
(
t
,
r
,
μ
,
N
)
∥
f
∥
L
t
∥
h
∥
L
r
.
If t=r=2N2N-μ, then
C
(
t
,
r
,
μ
,
N
)
=
C
(
N
,
μ
)
=
π
μ
2
Γ
(
N
2
-
μ
2
)
Γ
(
N
-
μ
2
)
{
Γ
(
N
2
)
Γ
(
μ
2
)
}
-
1
+
μ
N
.
Equality holds in (2.1) if and only if fh≡constant and
h
(
x
)
=
A
(
γ
2
+
|
x
-
a
|
2
)
2
N
-
μ
2
for some A∈C,0≠γ∈R and a∈RN.
We consider the following functional space:
D
1
,
2
(
ℝ
N
)
:=
{
u
∈
L
2
*
(
ℝ
N
)
:
∇
u
∈
L
2
(
ℝ
N
)
}
,
endowed with the norm defined as
∥
u
∥
:=
(
∫
ℝ
N
|
∇
u
|
2
𝑑
x
)
1
2
.
The space D01,2(Ω) is defined as the closure of Cc∞(Ω) in D1,2(ℝN).
Definition 2.2.
A function u∈D01,2(Ω) is said to be a solution of (P) if u satisfies
∫
Ω
∇
u
∇
ϕ
d
x
+
∫
Ω
∫
Ω
|
u
(
x
)
|
2
μ
*
|
u
(
y
)
|
2
μ
*
-
2
u
(
y
)
ϕ
(
y
)
|
x
-
y
|
μ
𝑑
x
𝑑
y
for all
ϕ
∈
D
0
1
,
2
(
Ω
)
.
Notation.
We define u+=max(u,0) and u-=max(-u,0) for all u∈D1,2(ℝN). Moreover, we set
ℝ
+
N
:=
{
x
∈
ℝ
N
:
x
N
>
0
}
and we denote by * the standard convolution operator.
Consider functionals I:D01,2(Ω)→ℝ and I∞:D1,2(ℝN)→ℝ given by
I
(
u
)
=
1
2
∫
Ω
|
∇
u
|
2
𝑑
x
-
1
2
⋅
2
μ
*
∫
Ω
∫
Ω
|
u
+
(
x
)
|
2
μ
*
|
u
+
(
y
)
|
2
μ
*
|
x
-
y
|
μ
𝑑
x
𝑑
y
,
u
∈
D
0
1
,
2
(
Ω
)
,
I
∞
(
u
)
=
1
2
∫
ℝ
N
|
∇
u
|
2
𝑑
x
1
2
⋅
2
μ
*
∫
ℝ
N
∫
ℝ
N
|
u
+
(
x
)
|
2
μ
*
|
u
+
(
y
)
|
2
μ
*
|
x
-
y
|
μ
𝑑
x
𝑑
y
,
u
∈
D
1
,
2
(
ℝ
N
)
.
By the Hardy–Littlewood–Sobolev inequality, we have
(
∫
ℝ
N
∫
ℝ
N
|
u
(
x
)
|
2
μ
*
|
u
(
y
)
|
2
μ
*
|
x
-
y
|
μ
𝑑
x
𝑑
y
)
1
2
μ
*
≤
C
(
N
,
μ
)
2
N
-
μ
N
-
2
∥
u
∥
L
2
*
2
,
where 2*=2NN-2. This implies that I∈C1(D01,2(Ω),ℝ) and I∞∈C1(D1,2(ℝN),ℝ). The best constant for the embedding D1,2(ℝN) into L2*(ℝN) is defined as
(2.2)
S
=
inf
u
∈
D
1
,
2
(
ℝ
N
)
∖
{
0
}
{
∫
ℝ
N
|
∇
u
|
2
𝑑
x
:
∫
R
N
|
u
|
2
*
𝑑
x
=
1
}
.
Consequently, we define
(2.3)
S
H
,
L
=
inf
u
∈
D
1
,
2
(
ℝ
N
)
∖
{
0
}
{
∫
ℝ
N
|
∇
u
|
2
𝑑
x
:
∫
ℝ
N
∫
ℝ
N
|
u
(
x
)
|
2
μ
*
|
u
(
y
)
|
2
μ
*
|
x
-
y
|
μ
𝑑
x
𝑑
y
=
1
}
.
It was established by Talenti [38] that the best constant S is achieved if and only if u is of the form
(
t
t
2
+
|
x
-
(
1
-
t
)
σ
|
2
)
N
-
2
2
for
σ
∈
Σ
:=
{
x
∈
ℝ
N
:
|
x
|
=
1
}
and
t
∈
(
0
,
1
]
.
Properties of the best constant SH,L were established by Gao and Yang [16]. We recall the following property.
Lemma 2.3.
The constant SH,L defined in (2.3) is achieved if and only if
u
=
C
(
b
b
2
+
|
x
-
a
|
2
)
N
-
2
2
,
where C>0 is a fixed constant, a∈RN and b∈(0,∞) are parameters. Moreover,
S
H
,
L
=
S
C
(
N
,
μ
)
N
-
2
2
N
-
μ
,
where S is defined as in (2.2).
The following property was established in [16].
Lemma 2.4.
If N≥3 and 0<μ<N, then
∥
⋅
∥
N
L
:=
(
∫
ℝ
N
∫
ℝ
N
|
⋅
|
2
μ
*
|
⋅
|
2
μ
*
|
x
-
y
|
μ
d
x
d
y
)
1
2
⋅
2
μ
*
defines a norm on L2*(RN).
Remark 2.5.
If we define
S
A
=
inf
u
∈
D
1
,
2
(
ℝ
N
)
∖
{
0
}
{
∫
ℝ
N
|
∇
u
|
2
𝑑
x
:
∫
ℝ
N
∫
ℝ
N
|
u
+
(
x
)
|
2
μ
*
|
u
+
(
y
)
|
2
μ
*
|
x
-
y
|
μ
𝑑
x
𝑑
y
=
1
}
,
then SA=SH,L.
Proposition 2.6.
Let u∈D01,2(Ω) be an arbitrary solution of the problem
(2.4)
{
-
Δ
u
=
(
∫
Ω
|
u
+
(
y
)
|
2
μ
*
|
x
-
y
|
μ
𝑑
y
)
|
u
+
|
2
μ
*
-
1
in
Ω
,
u
=
0
on
∂
Ω
.
Then
I
(
u
)
≥
1
2
(
N
-
μ
+
2
2
N
-
μ
)
S
H
,
L
2
N
-
μ
N
-
μ
+
2
=
:
β
.
Moreover, the same conclusion holds for the solution u∈D1,2(RN) of
-
Δ
u
=
(
∫
ℝ
N
|
u
+
(
y
)
|
2
μ
*
|
x
-
y
|
μ
𝑑
y
)
|
u
+
|
2
μ
*
-
1
in
ℝ
N
.
Proof.
If u is a solution of (2.4), then testing (2.4) with u+ and u- yields
∫
Ω
|
∇
u
+
|
2
𝑑
x
=
∫
Ω
∫
Ω
|
u
+
(
x
)
|
2
μ
*
|
u
+
(
y
)
|
2
μ
*
|
x
-
y
|
μ
𝑑
x
𝑑
y
and
∫
Ω
|
∇
u
-
|
2
𝑑
x
=
0
a.e. on
Ω
.
It follows that
(
S
A
)
2
μ
*
2
μ
*
-
1
≤
∫
Ω
∫
Ω
|
u
+
(
x
)
|
2
μ
*
|
u
+
(
y
)
|
2
μ
*
|
x
-
y
|
μ
𝑑
x
𝑑
y
=
∫
Ω
|
∇
u
+
|
2
𝑑
x
=
∫
Ω
|
∇
u
|
2
𝑑
x
.
We obtain
I
(
u
)
≥
(
1
2
-
1
2
⋅
2
μ
*
)
(
S
A
)
2
μ
*
2
μ
*
-
1
=
1
2
(
N
-
μ
+
2
2
N
-
μ
)
S
H
,
L
2
N
-
μ
N
-
μ
+
2
.
The proof is now complete. ∎
Lemma 2.7 (Pohozaev Identity).
Let N≥3 and assume that u∈D01,2(R+N) solves
(2.5)
-
Δ
u
=
(
∫
ℝ
+
N
|
u
+
(
y
)
|
2
μ
*
|
x
-
y
|
μ
𝑑
y
)
|
u
+
|
2
μ
*
-
1
in
ℝ
+
N
.
Then the following equality holds:
1
2
∫
∂
ℝ
+
N
(
x
-
x
0
)
⋅
ν
|
∇
u
|
2
𝑑
S
+
N
-
2
2
∫
ℝ
+
N
|
∇
u
|
2
𝑑
x
=
2
N
-
μ
2
⋅
2
μ
*
∫
ℝ
+
N
∫
ℝ
+
N
|
u
+
(
x
)
|
2
μ
*
|
u
+
(
y
)
|
2
μ
*
|
x
-
y
|
μ
𝑑
x
𝑑
y
,
where ν is the unit outward normal to ∂Ω and x0=(0,0,…,1).
Proof.
First observe that any solution of problem (2.5) is nonnegative. This implies
∇
u
=
∇
u
+
a.e. on
ℝ
+
N
.
Extending u=0 in ℝN∖ℝ+N, we have u∈Wloc2,2(ℝN) (see Lemma 3.1). Now fix φ∈Cc1(ℝN) such that φ=1 on B1. Let the function φλ∈D1,2(ℝN) be defined for λ∈(0,∞) and x∈ℝN by φλ(x)=φ(λx). Multiplying (2.5) with ((x-x0)⋅∇u)φλ and integrating over ℝ+N, we obtain
(2.6)
∫
ℝ
+
N
(
-
Δ
u
)
(
(
x
-
x
0
)
⋅
∇
u
)
φ
λ
(
x
)
𝑑
x
=
∫
ℝ
+
N
(
∫
ℝ
+
N
|
u
+
(
y
)
|
2
μ
*
|
x
-
y
|
μ
𝑑
y
)
|
u
+
|
2
μ
*
-
1
(
(
x
-
x
0
)
⋅
∇
u
)
φ
λ
𝑑
x
=
∫
ℝ
+
N
∇
(
(
x
-
x
0
)
∫
ℝ
+
N
(
|
u
+
(
y
)
|
2
μ
*
|
x
-
y
|
μ
d
y
)
|
u
+
(
x
)
|
2
μ
*
-
1
φ
λ
(
x
)
u
(
x
)
)
d
x
-
∫
ℝ
+
N
u
(
x
)
∇
(
(
x
-
x
0
)
∫
ℝ
+
N
(
|
u
+
(
y
)
|
2
μ
*
|
x
-
y
|
μ
d
y
)
|
u
+
(
x
)
|
2
μ
*
-
1
φ
λ
(
x
)
)
d
x
.
Using the divergence theorem on the right-hand side of (2.6), we obtain
(2.7)
∫
ℝ
+
N
(
-
Δ
u
)
(
(
x
-
x
0
)
⋅
∇
u
)
φ
λ
(
x
)
𝑑
x
=
∫
ℝ
+
N
(
∫
ℝ
+
N
|
u
+
(
y
)
|
2
μ
*
|
x
-
y
|
μ
𝑑
y
)
|
u
+
|
2
μ
*
-
1
(
(
x
-
x
0
)
⋅
∇
u
)
φ
λ
𝑑
x
=
-
∫
ℝ
+
N
u
(
x
)
∇
(
(
x
-
x
0
)
∫
ℝ
+
N
(
|
u
+
(
y
)
|
2
μ
*
|
x
-
y
|
μ
d
y
)
|
u
+
(
x
)
|
2
μ
*
-
1
φ
λ
(
x
)
)
d
x
.
Now consider the integral
(2.8)
∫
ℝ
+
N
u
(
x
)
∇
(
(
x
-
x
0
)
∫
ℝ
+
N
(
|
u
+
(
y
)
|
2
μ
*
|
x
-
y
|
μ
d
y
)
|
u
+
(
x
)
|
2
μ
*
-
1
φ
λ
(
x
)
)
d
x
=
∫
ℝ
+
N
N
u
(
x
)
(
∫
ℝ
+
N
|
u
+
(
y
)
|
2
μ
*
|
x
-
y
|
μ
𝑑
y
)
|
u
+
(
x
)
|
2
μ
*
-
1
φ
λ
(
x
)
𝑑
x
+
∫
ℝ
+
N
(
2
μ
*
-
1
)
u
(
x
)
(
∫
ℝ
+
N
|
u
+
(
y
)
|
2
μ
*
|
x
-
y
|
μ
𝑑
y
)
|
u
+
(
x
)
|
2
μ
*
-
2
φ
λ
(
x
)
(
∇
u
⋅
(
x
-
x
0
)
)
𝑑
x
-
μ
∫
ℝ
+
N
u
(
x
)
φ
λ
(
x
)
(
∫
ℝ
+
N
|
u
+
(
y
)
|
2
μ
*
(
x
-
x
0
)
⋅
(
x
-
y
)
|
x
-
y
|
μ
+
2
𝑑
y
)
|
u
+
(
x
)
|
2
μ
*
-
1
𝑑
x
+
λ
∫
ℝ
+
N
∫
ℝ
+
N
|
u
+
(
y
)
|
2
μ
*
|
u
+
(
x
)
|
2
μ
*
|
x
-
y
|
μ
(
x
-
x
0
)
⋅
∇
φ
(
λ
x
)
𝑑
x
𝑑
y
.
Taking into account (2.7) and (2.8), we have
(2.9)
2
μ
*
∫
ℝ
+
N
(
x
-
x
0
)
⋅
∇
u
(
x
)
(
∫
ℝ
+
N
|
u
+
(
y
)
|
2
μ
*
|
x
-
y
|
μ
𝑑
y
)
|
u
+
(
x
)
|
2
μ
*
-
1
φ
λ
(
x
)
𝑑
x
=
-
N
∫
ℝ
+
N
u
(
x
)
(
∫
ℝ
+
N
|
u
+
(
y
)
|
2
μ
*
|
x
-
y
|
μ
𝑑
y
)
|
u
+
(
x
)
|
2
μ
*
-
1
φ
λ
(
x
)
𝑑
x
+
μ
∫
ℝ
+
N
u
(
x
)
φ
λ
(
x
)
(
∫
ℝ
+
N
|
u
+
(
y
)
|
2
μ
*
(
x
-
x
0
)
.
(
x
-
y
)
|
x
-
y
|
μ
+
2
𝑑
y
)
|
u
+
(
x
)
|
2
μ
*
-
1
𝑑
x
-
λ
∫
ℝ
+
N
∫
ℝ
+
N
|
u
+
(
y
)
|
2
μ
*
|
u
+
(
x
)
|
2
μ
*
|
x
-
y
|
μ
(
x
-
x
0
)
⋅
∇
φ
(
λ
x
)
𝑑
x
𝑑
y
.
Now, interchanging the role of x and y in (2.9) and combining the resultant equation with (2.9), we deduce that
∫
ℝ
+
N
(
x
-
x
0
)
⋅
∇
u
(
x
)
∫
ℝ
+
N
(
|
u
+
(
y
)
|
2
μ
*
|
x
-
y
|
μ
d
y
)
|
u
+
(
x
)
|
2
μ
*
-
1
φ
λ
(
x
)
𝑑
x
=
μ
-
2
N
2
⋅
2
μ
*
∫
ℝ
+
N
∫
ℝ
+
N
|
u
+
(
y
)
|
2
μ
*
|
u
+
(
x
)
|
2
μ
*
|
x
-
y
|
μ
φ
λ
(
x
)
𝑑
x
𝑑
y
-
λ
2
μ
*
∫
ℝ
+
N
∫
ℝ
+
N
|
u
+
(
y
)
|
2
μ
*
|
u
+
(
x
)
|
2
μ
*
|
x
-
y
|
μ
(
x
-
x
0
)
⋅
∇
φ
(
λ
x
)
𝑑
x
𝑑
y
.
Passing to the limit as λ→0 and using the dominated convergence theorem, we obtain that
(2.10)
∫
ℝ
+
N
(
x
-
x
0
)
⋅
∇
u
(
x
)
(
∫
ℝ
+
N
|
u
+
(
y
)
|
2
μ
*
|
x
-
y
|
μ
𝑑
y
)
|
u
+
(
x
)
|
2
μ
*
-
1
𝑑
x
=
μ
-
2
N
2
⋅
2
μ
*
∫
ℝ
+
N
∫
ℝ
+
N
|
u
+
(
y
)
|
2
μ
*
|
u
+
(
x
)
|
2
μ
*
|
x
-
y
|
μ
𝑑
x
𝑑
y
.
It is easily seen that
Δ
u
(
(
x
-
x
0
)
⋅
∇
u
)
φ
λ
=
div
(
∇
u
φ
λ
(
x
-
x
0
)
⋅
∇
u
)
-
φ
λ
|
∇
u
|
2
-
φ
λ
(
x
-
x
0
)
⋅
∇
(
|
∇
u
|
2
2
)
-
λ
(
(
x
-
x
0
)
⋅
∇
u
)
(
∇
φ
(
λ
x
)
⋅
∇
u
)
=
div
(
(
∇
u
(
x
-
x
0
)
⋅
∇
u
-
(
x
-
x
0
)
|
∇
u
|
2
2
)
φ
λ
)
+
N
-
2
2
φ
λ
|
∇
u
|
2
+
λ
|
∇
u
|
2
2
(
(
x
-
x
0
)
⋅
∇
φ
(
λ
x
)
)
-
λ
(
(
x
-
x
0
)
⋅
∇
u
)
(
∇
φ
(
λ
x
)
⋅
∇
u
)
.
Now, integrating by parts we obtain
∫
ℝ
+
N
(
Δ
u
)
(
(
x
-
x
0
)
⋅
∇
u
)
φ
λ
𝑑
x
=
∫
∂
ℝ
+
N
(
∇
u
(
x
-
x
0
)
⋅
∇
u
-
(
x
-
x
0
)
|
∇
u
|
2
2
)
φ
λ
⋅
ν
𝑑
S
+
N
-
2
2
∫
ℝ
+
N
φ
λ
|
∇
u
|
2
𝑑
x
-
∫
ℝ
+
N
λ
|
∇
u
|
2
2
(
(
x
-
x
0
)
⋅
∇
φ
(
λ
x
)
)
𝑑
x
-
∫
ℝ
+
N
λ
(
(
x
-
x
0
)
⋅
∇
u
)
(
∇
φ
(
λ
x
)
⋅
∇
u
)
𝑑
x
.
Noticing that ∇u=(∇u⋅ν)ν on ∂ℝ+N and employing dominated convergence theorem for λ→0, we get that
(2.11)
∫
ℝ
+
N
(
Δ
u
)
(
(
x
-
x
0
)
⋅
∇
u
)
=
1
2
∫
∂
ℝ
+
N
|
∇
u
|
2
(
x
-
x
0
)
⋅
ν
𝑑
S
+
N
-
2
2
∫
ℝ
+
N
|
∇
u
|
2
𝑑
x
.
From equation (2.6), (2.10) and (2.11) we have our desired result. ∎
We can now deduce the following Liouville-type theorem.
Theorem 2.8.
Let N≥3 and let u∈D01,2(R+N) be any solution of
(2.12)
-
Δ
u
=
(
∫
ℝ
+
N
|
u
+
(
y
)
|
2
μ
*
|
x
-
y
|
μ
𝑑
y
)
|
u
+
|
2
μ
*
-
1
in
ℝ
+
N
.
Then u≡0 on R+N.
Proof.
If u is a solution of (2.12), then
∫
ℝ
+
N
∇
u
⋅
∇
ϕ
d
x
-
∫
ℝ
+
N
∫
ℝ
+
N
|
u
+
(
x
)
|
2
μ
*
|
u
+
(
y
)
|
2
μ
*
-
1
ϕ
(
y
)
|
x
-
y
|
μ
𝑑
x
𝑑
y
for all
ϕ
∈
D
0
1
,
2
(
ℝ
+
N
)
.
Taking ϕ=u- we obtain u-=0 a.e. on ℝN. This implies that u is a nonnegative solution of (2.12). Now, by Lemma 2.7 we have
∫
∂
ℝ
+
N
|
∇
u
|
2
(
x
-
x
0
)
⋅
ν
𝑑
S
=
0
.
But (x-x0)⋅ν>0 for x∈∂ℝ+N. Since u is a nontrivial solution, we get a contradiction from the Hopf boundary point lemma. Hence, u≡0 on ℝ+N. ∎
3 Classification of Solutions
In this section we will discuss the regularity and classification of nonnegative solutions of the following equation:
(3.1)
-
Δ
u
=
(
|
x
|
μ
-
N
*
|
u
|
p
)
|
u
|
p
-
2
u
in
ℝ
N
,
where p:=N+μN-2 and 0<μ<N. Consider the following integral system of equations:
(3.2)
{
u
(
x
)
=
∫
ℝ
N
u
p
-
1
(
y
)
v
(
y
)
|
x
-
y
|
N
-
2
𝑑
y
,
u
≥
0
in
ℝ
N
,
v
(
x
)
=
∫
ℝ
N
u
p
(
y
)
|
x
-
y
|
N
-
μ
𝑑
y
,
v
≥
0
in
ℝ
N
.
We note that if u∈D1,2(ℝN), then (u,v)∈L2NN-2(ℝN)×L2NN-μ(ℝN).
First we will discuss the regularity of nonnegative solutions of (3.1). In this regard, we will prove the following auxiliary result.
Lemma 3.1.
Let u∈D1,2(RN) be a nonnegative solutions of (3.1). Then u∈Wloc2,s(RN) for all 1≤s<∞.
Proof.
Let u∈D1,2(ℝN) be a nonnegative solution of (3.1). Now following the same approach as in proof of [18, Lemma 3.1], we have (u,v)∈Lr(ℝN)×Ls(ℝN) for all 1<r,s<∞. In particular, up∈LNμ(ℝN), and now using the boundedness of Riesz potential operator, we have |x|μ-N*up∈L∞(ℝN). Thus, from (3.1), we have
|
-
Δ
u
|
≤
C
|
u
|
p
-
1
.
By the classical elliptic regularity theory for subcritical problems in local bounded domains, we have u∈Wloc2,s(ℝN) for any 1≤s<∞. ∎
Next, we will discuss the classification of all positive solutions of the following system:
(3.3)
{
u
(
x
)
=
∫
ℝ
N
u
a
(
y
)
v
b
(
y
)
|
x
-
y
|
N
-
α
𝑑
y
,
u
>
0
in
ℝ
N
,
v
(
x
)
=
∫
ℝ
N
u
c
(
y
)
v
d
(
y
)
|
x
-
y
|
N
-
β
𝑑
y
,
v
>
0
in
ℝ
N
,
where a≥0, b,c,d∈{0}∪[1,∞), 0<α,β<N.
Let (u,v)∈Lq1(ℝN)×Lq2(ℝN) be a solution of (3.3). Now for all λ∈ℝ, we define
T
λ
:=
{
(
x
1
,
x
2
,
…
,
x
n
)
∈
ℝ
N
:
x
1
=
λ
}
as the moving plane. Let
x
λ
:=
(
2
λ
-
x
1
,
x
2
,
…
,
x
n
)
,
let
Σ
λ
:=
{
(
x
1
,
x
2
,
…
,
x
n
)
∈
ℝ
N
:
x
1
<
λ
}
and let
Σ
λ
′
:=
{
(
x
1
,
x
2
,
…
,
x
n
)
∈
ℝ
N
:
x
1
≥
λ
}
be the reflection of Σλ about the plane Tλ. Moreover, define uλ(y):=u(yλ) and vλ(y)=v(yλ). Immediately, we have the following property whose proof is just an elementary computation.
Lemma 3.2.
Assume that (u,v) is a positive pair of solution of (3.3). Then
{
u
(
y
λ
)
-
u
(
y
)
=
∫
Σ
λ
(
1
|
y
-
x
|
N
-
α
-
1
|
y
α
-
x
|
N
-
α
)
[
u
a
(
x
λ
)
v
b
(
x
λ
)
-
u
a
(
x
)
v
b
(
x
)
]
𝑑
x
,
v
(
y
λ
)
-
v
(
y
)
=
∫
Σ
λ
(
1
|
y
-
x
|
N
-
β
-
1
|
y
α
-
x
|
N
-
β
)
[
u
c
(
x
λ
)
v
d
(
x
λ
)
-
u
c
(
x
)
v
d
(
x
)
]
𝑑
x
.
Lemma 3.3.
There exists η>0 such that for all λ<-η,
u
(
y
λ
)
≥
u
(
y
)
,
v
(
y
λ
)
≥
v
(
y
)
for all
y
∈
Σ
λ
.
Proof.
Define Σλu:={y∈Σλ:u(y)>uλ(y)}, Σλv:={y∈Σλ:v(y)>vλ(y)}. By Lemma 3.2, we obtain
u
(
y
λ
)
-
u
(
y
)
=
∫
Σ
λ
(
1
|
y
-
x
|
N
-
α
-
1
|
y
λ
-
x
|
N
-
α
)
[
u
a
(
x
λ
)
v
b
(
x
λ
)
-
u
a
(
x
)
v
b
(
x
)
]
𝑑
x
≤
∫
Σ
λ
(
1
|
y
-
x
|
N
-
α
-
1
|
y
λ
-
x
|
N
-
α
)
[
u
λ
a
(
v
b
-
v
λ
b
)
+
+
v
b
(
u
a
-
u
λ
a
)
+
]
𝑑
x
≤
∫
Σ
λ
1
|
y
-
x
|
N
-
α
[
u
λ
a
(
v
b
-
v
λ
b
)
+
+
v
b
(
u
a
-
u
λ
a
)
+
]
𝑑
x
.
By the Hardy–Littlewood–Sobolev inequality, we obtain
∥
u
-
u
λ
∥
L
q
1
(
Σ
λ
u
)
≤
∥
u
-
u
λ
∥
L
q
1
(
Σ
λ
)
≤
C
∥
u
λ
a
(
v
b
-
v
λ
b
)
+
+
v
b
(
u
a
-
u
λ
a
)
+
∥
L
r
(
Σ
λ
)
≤
C
∥
u
λ
a
(
v
b
-
v
λ
b
)
∥
L
r
(
Σ
λ
v
)
+
∥
v
b
(
u
a
-
u
λ
a
)
∥
L
r
(
Σ
λ
u
)
,
where r=Nq1N+αq1. Now if a,b>1, then by Hölder’s inequality, we get
(3.4)
∥
u
-
u
λ
∥
L
q
1
(
Σ
λ
u
)
≤
C
∥
u
λ
a
v
b
-
1
(
v
-
v
λ
)
∥
L
r
(
Σ
λ
v
)
+
C
∥
v
b
u
a
-
1
(
u
-
u
λ
)
∥
L
r
(
Σ
λ
u
)
≤
C
∥
u
λ
∥
L
q
1
(
Σ
λ
v
)
a
∥
v
b
-
1
(
v
-
v
λ
)
∥
L
s
(
Σ
λ
v
)
+
C
∥
v
∥
L
q
2
(
Σ
λ
u
)
b
∥
u
a
-
1
(
u
-
u
λ
)
∥
L
t
(
Σ
λ
u
)
≤
C
∥
u
λ
∥
L
q
1
(
Σ
λ
′
)
a
∥
v
∥
L
q
2
(
Σ
λ
v
)
b
-
1
∥
v
-
v
λ
∥
L
q
2
(
Σ
λ
v
)
+
C
∥
v
∥
L
q
2
(
Σ
λ
)
b
∥
u
∥
L
q
1
(
Σ
λ
u
)
a
-
1
∥
u
-
u
λ
∥
L
q
1
(
Σ
λ
u
)
,
and if 0<a<1, b>1, then we have
(3.5)
∥
u
-
u
λ
∥
L
q
1
(
Σ
λ
u
)
≤
C
∥
u
λ
a
v
b
-
1
(
v
-
v
λ
)
∥
L
r
(
Σ
λ
v
)
+
C
∥
v
b
(
u
-
u
λ
)
a
∥
L
r
(
Σ
λ
u
)
≤
C
∥
u
λ
∥
L
q
1
(
Σ
λ
v
)
a
∥
v
b
-
1
(
v
-
v
λ
)
∥
L
s
(
Σ
λ
v
)
+
C
∥
v
∥
L
q
2
(
Σ
λ
u
)
b
∥
u
-
u
λ
∥
L
q
1
(
Σ
λ
u
)
a
≤
C
∥
u
λ
∥
L
q
1
(
Σ
λ
′
)
a
∥
v
∥
L
q
2
(
Σ
λ
v
)
b
-
1
∥
v
-
v
λ
∥
L
q
2
(
Σ
λ
v
)
+
C
∥
v
∥
L
q
2
(
Σ
λ
)
b
∥
u
-
u
λ
∥
L
q
1
(
Σ
λ
u
)
,
where
s
=
r
q
1
q
1
-
a
r
,
t
=
r
q
2
q
2
-
b
r
=
q
1
r
and
b
q
2
+
a
-
1
q
1
=
α
N
.
Similarly, for c,d>1 we have
(3.6)
∥
v
-
v
λ
∥
L
q
2
(
Σ
λ
v
)
≤
C
∥
v
∥
L
q
2
(
Σ
λ
′
)
d
∥
u
∥
L
q
1
(
Σ
λ
u
)
c
-
1
∥
u
-
u
λ
∥
L
q
1
(
Σ
λ
u
)
+
C
∥
u
∥
L
q
1
(
Σ
λ
)
c
∥
v
∥
L
q
2
(
Σ
λ
v
)
d
-
1
∥
v
-
v
λ
∥
L
q
2
(
Σ
λ
v
)
,
where q1 and q2 are positive constant such that d-1q2+cq1=βN. Taking into account (3.4), (3.5) and (3.6), for all λ∈ℝ we have
∥
u
-
u
λ
∥
L
q
1
(
Σ
λ
u
)
≤
{
C
∥
v
∥
L
q
2
(
Σ
λ
′
)
d
∥
u
∥
L
q
1
(
Σ
λ
u
)
c
-
1
1
-
C
∥
u
∥
L
q
1
(
Σ
λ
)
c
∥
v
∥
L
q
2
(
Σ
λ
v
)
d
-
1
∥
u
λ
∥
L
q
1
(
Σ
λ
′
)
a
∥
v
∥
L
q
2
(
Σ
λ
v
)
b
-
1
+
C
∥
v
∥
L
q
2
(
Σ
λ
)
b
∥
u
∥
L
q
1
(
Σ
λ
u
)
a
-
1
}
∥
u
-
u
λ
∥
L
q
1
(
Σ
λ
u
)
.
Using the fact that (u,v)∈Lq1(ℝN)×Lq2(ℝN), we can choose η>0 sufficiently large such that for all λ<-η.
C
∥
v
∥
L
q
2
(
Σ
λ
′
)
d
∥
u
∥
L
q
1
(
Σ
λ
u
)
c
-
1
1
-
C
∥
u
∥
L
q
1
(
Σ
λ
)
c
∥
v
∥
L
q
2
(
Σ
λ
v
)
d
-
1
∥
u
λ
∥
L
q
1
(
Σ
λ
′
)
a
∥
v
∥
L
q
2
(
Σ
λ
v
)
b
-
1
+
C
∥
v
∥
L
q
2
(
Σ
λ
)
b
∥
u
∥
L
q
1
(
Σ
λ
u
)
a
-
1
≤
1
2
.
It follows that ∥u-uλ∥Lq1(Σλu)=0 and hence Σλu must be measure zero and empty when λ<-η. In the similar manner, Σλv must be of measure zero and empty when λ<-η. For all other cases, the proof follows analogously. This concludes the proof of the lemma. ∎
Now using the same assertions and arguments as in Huang, Li and Wang [18] in combination with Lemma 3.3, we have the following theorem.
Theorem 3.4.
Assume that a≥0, b,c,d∈{0}∪[1,∞), 0<α,β<N and (u,v)∈Lq1(RN)×Lq2(RN) is a pair of positive solutions of (3.3) with q1 and q2 satisfying
q
1
,
q
2
>
1
,
b
q
2
+
a
-
1
q
1
=
α
N
,
c
q
1
+
d
-
1
q
2
=
β
N
.
Then (u,v) is radially symmetric and monotone decreasing about some point in RN. Moreover, if
b
=
1
N
-
β
[
(
N
+
α
)
-
a
(
N
-
α
)
]
,
c
=
1
N
-
α
[
(
N
+
β
)
-
d
(
N
-
β
)
]
,
then (u,v) must be of the form
u
(
x
)
=
(
d
1
e
1
+
|
x
-
x
1
|
2
)
N
-
α
2
,
v
(
x
)
=
(
d
2
e
2
+
|
x
-
x
2
|
2
)
N
-
β
2
for some constants d1,d2,e1,e2>0 and some x1,x2∈RN.
As an immediate corollary, we have the following result on radial symmetry of nonnegative solutions of (3.1).
Corollary 3.5.
Every nonnegative solution u∈D1,2(RN) of equation (3.1) is radially symmetric, monotone decreasing and of the form
u
(
x
)
=
(
c
1
c
2
+
|
x
-
x
0
|
2
)
N
-
2
2
for some constants c1,c2>0 and some x0∈RN.
Proof.
Let u be any nonnegative solution of equation (3.1). Then by Lemma 3.1, we have u∈Wloc2,s(ℝN) for any 1≤s<∞. Hence, by the strong maximum principle, we have that u is a positive function in ℝN. It implies that (u,v)∈L2NN-2(ℝN)×L2NN-μ(ℝN) is a positive solution of the integral system (3.2). Now employing Theorem 3.4 for α=2, a=p-1, b=1, β=μ, c=p, d=0 and using the fact u∈D1,2(ℝN), that is, u∈L2NN-2(ℝN) and v∈L2NN-μ(ℝN), we have the desired result. ∎
4 Palais–Smale Analysis
Lemma 4.1.
Let un⇀u be weakly convergent in D1,2(RN) and un→u a.e. on RN. Then
(4.1)
(
|
x
|
-
μ
*
|
(
u
n
)
+
|
2
μ
*
)
|
(
u
n
)
+
|
2
μ
*
-
2
(
u
n
)
+
-
(
|
x
|
-
μ
*
|
(
u
n
-
u
)
+
|
2
μ
*
)
|
(
u
n
-
u
)
+
|
2
μ
*
-
2
(
u
n
-
u
)
+
→
(
|
x
|
-
μ
*
|
u
+
|
2
μ
*
)
|
u
+
|
2
μ
*
-
2
u
+
in
(
D
1
,
2
(
ℝ
N
)
)
′
.
Proof.
Since un⇀u weakly in D1,2(ℝN), there exists M>0 such that ∥un∥<M for all n∈ℕ. Let ϕ∈D1,2(ℝN) and
I
=
∫
ℝ
N
[
(
|
x
|
-
μ
*
|
(
u
n
)
+
|
2
μ
*
)
|
(
u
n
)
+
|
2
μ
*
-
2
(
u
n
)
+
(
|
x
|
-
μ
*
|
(
u
n
-
u
)
+
|
2
μ
*
)
|
(
u
n
-
u
)
+
|
2
μ
*
-
2
(
u
n
-
u
)
+
]
ϕ
𝑑
x
.
Then I=I1+I2+I3-2I4, where
I
1
=
∫
ℝ
N
(
|
x
|
-
μ
*
(
|
(
u
n
)
+
|
2
μ
*
-
|
(
u
n
-
u
)
+
|
2
μ
*
)
)
(
|
(
u
n
)
+
|
2
μ
*
-
2
(
u
n
)
+
-
|
(
u
n
-
u
)
+
|
2
μ
*
-
2
(
u
n
-
u
)
+
)
ϕ
𝑑
x
,
I
2
=
∫
ℝ
N
(
|
x
|
-
μ
*
|
(
u
n
)
+
|
2
μ
*
)
|
(
u
n
-
u
)
+
|
2
μ
*
-
2
(
u
n
-
u
)
+
ϕ
𝑑
x
,
I
3
=
∫
ℝ
N
(
|
x
|
-
μ
*
|
(
u
n
-
u
)
+
|
2
μ
*
)
|
(
u
n
)
+
|
2
μ
*
-
2
(
u
n
)
+
ϕ
𝑑
x
,
I
4
=
∫
ℝ
N
(
|
x
|
-
μ
*
|
(
u
n
-
u
)
+
|
2
μ
*
)
|
(
u
n
-
u
)
+
|
2
μ
*
-
2
(
u
n
-
u
)
+
ϕ
𝑑
x
.
Claim 1.
We have
lim
n
→
∞
I
1
=
∫
ℝ
N
(
|
x
|
-
μ
*
|
u
+
|
2
μ
*
)
|
u
+
|
2
μ
*
-
2
u
+
ϕ
𝑑
x
.
Similar to the proof of the Brezis–Lieb lemma [8] we have
|
(
u
n
)
+
|
2
μ
*
-
|
(
u
n
-
u
)
+
|
2
μ
*
→
|
u
+
|
2
μ
*
in
L
2
N
2
N
-
μ
(
ℝ
N
)
as
n
→
∞
.
Since the Hardy–Littlewood–Sobolev inequality implies that the Riesz potential defines a linear continuous map from L2N2N-μ(ℝN) to L2Nμ(ℝN), we get
(4.2)
|
x
|
-
μ
*
(
|
(
u
n
)
+
|
2
μ
*
-
|
(
u
n
-
u
)
+
|
2
μ
*
)
→
|
x
|
-
μ
*
|
u
+
|
2
μ
*
strongly in
L
2
N
μ
(
ℝ
N
)
as
n
→
∞
.
Since both |(un)+|2μ*-2(un)+ϕ⇀|u+|2μ*-2u+ϕ and |(un-u)+|2μ*-2(un-u)+ϕ⇀0 converge weakly in L2N2N-μ(ℝN), we obtain
(4.3)
|
(
u
n
)
+
|
2
μ
*
-
2
(
u
n
)
+
ϕ
-
|
(
u
n
-
u
)
+
|
2
μ
*
-
2
(
u
n
-
u
)
+
ϕ
⇀
|
u
+
|
2
μ
*
-
2
u
+
ϕ
weakly in L2N2N-μ(ℝN). Thus, Claim 1 follows from (4.2) and (4.3).
Claim 2.
We have limn→∞I2=0.
Since |(un)+|2μ*⇀|(u)+|2μ* weakly in L2N2N-μ(ℝN), by the Hardy–Littlewood–Sobolev inequality (2.1) we have
(4.4)
|
x
|
-
μ
*
|
(
u
n
)
+
|
2
μ
*
⇀
|
x
|
-
μ
*
|
u
+
|
2
μ
*
weakly in
L
2
N
μ
(
ℝ
N
)
.
We observe that
|
(
u
n
-
u
)
+
|
2
μ
*
-
2
(
u
n
-
u
)
+
ϕ
→
0
a.e. in
ℝ
N
and for any open subset U⊂ℝN, we have
∫
U
|
|
(
u
n
-
u
)
+
|
2
μ
*
-
2
(
u
n
-
u
)
+
ϕ
|
2
N
2
N
-
μ
𝑑
x
≤
(
∫
U
|
(
u
n
-
u
)
+
|
2
*
𝑑
x
)
N
-
μ
+
2
2
N
-
μ
(
∫
U
|
ϕ
|
2
*
𝑑
x
)
N
-
2
2
N
-
μ
≤
∥
u
n
∥
2
*
(
2
μ
*
-
1
)
(
∫
U
|
ϕ
|
2
*
𝑑
x
)
N
-
2
2
N
-
μ
≤
M
(
∫
U
|
ϕ
|
2
*
𝑑
x
)
N
-
2
2
N
-
μ
.
This implies that {||(un-u)+|2μ*-2(un-u)+ϕ|2N2N-μ}n is equi-integrable in L1(ℝN). Hence, by the Vitali convergence theorem we get that
|
(
u
n
-
u
)
+
|
2
μ
*
-
2
(
u
n
-
u
)
+
ϕ
→
0
strongly in
L
2
N
2
N
-
μ
(
ℝ
N
)
.
This fact together with (4.4) completes the proof of Claim 2.
Claim 3.
We have limn→∞I3=0.
Similar to the proof of Claim 2, we have
|
x
|
-
μ
*
|
(
u
n
-
u
)
+
|
2
μ
*
⇀
0
weakly in
L
2
N
μ
(
ℝ
N
)
and
|
(
u
n
)
+
|
2
μ
*
-
2
(
u
n
)
+
ϕ
→
|
u
+
|
2
μ
*
-
2
u
+
ϕ
strongly in
L
2
N
2
N
-
μ
(
ℝ
N
)
.
Thus, Claim 3 follows.
Claim 4.
We have limn→∞I4=0.
Similar to the proof of Claim 2, we have
|
x
|
-
μ
*
|
(
u
n
-
u
)
+
|
2
μ
*
⇀
0
weakly in
L
2
N
μ
(
ℝ
N
)
and
|
(
u
n
-
u
)
+
|
2
μ
*
-
2
(
u
n
-
u
)
+
ϕ
→
0
strongly in
L
2
N
2
N
-
μ
(
ℝ
N
)
.
Thus, Claim 4 follows. Hence
I
→
∫
ℝ
N
(
|
x
|
-
μ
*
|
u
+
|
2
μ
*
)
|
u
+
|
2
μ
*
-
2
u
+
ϕ
𝑑
x
,
that is, (4.1) holds. ∎
Lemma 4.2.
If un⇀u weakly in D01,2(Ω), un→u a.e. on Ω, I(un)→c, I′(un)→0 in (D01,2(Ω))′, then I′(u)=0 and vn:=un-u satisfies
∥
v
n
∥
2
=
∥
u
n
∥
2
-
∥
u
∥
2
+
o
(
1
)
,
I
∞
(
v
n
)
→
c
-
I
(
u
)
𝑎𝑛𝑑
I
∞
′
(
v
n
)
→
0
in
(
D
0
1
,
2
(
Ω
)
)
′
.
Proof.
Let us prove the following:
Note that
u
n
⇀
u
weakly in
D
0
1
,
2
(
Ω
)
⟹
|
(
u
n
)
+
|
2
μ
*
⇀
|
u
+
|
2
μ
*
weakly in
L
2
N
2
N
-
μ
(
Ω
)
.
Since Riesz potential is a linear continuous map from L2N2N-μ(Ω) to L2Nμ(Ω), we obtain that
∫
Ω
|
(
u
n
)
+
(
y
)
|
2
μ
*
|
x
-
y
|
μ
𝑑
y
⇀
∫
Ω
|
u
+
(
y
)
|
2
μ
*
|
x
-
y
|
μ
𝑑
y
weakly in
L
2
N
μ
(
Ω
)
.
Also, |(un)+|2μ*-2(un)+⇀|u+|2μ*-2u+ weakly in L2NN-μ+2(Ω). Combining these facts, we have
(
∫
Ω
|
(
u
n
)
+
(
y
)
|
2
μ
*
|
x
-
y
|
μ
𝑑
y
)
|
(
u
n
)
+
|
2
μ
*
-
2
(
u
n
)
+
⇀
(
∫
Ω
|
u
+
(
y
)
|
2
μ
*
|
x
-
y
|
μ
𝑑
y
)
|
u
+
|
2
μ
*
-
2
u
+
weakly in
L
2
N
N
+
2
(
Ω
)
.
This implies for any ϕ∈D01,2(Ω), we have
(4.5)
∫
Ω
∫
Ω
|
(
u
n
)
+
(
x
)
|
2
μ
*
|
(
u
n
)
+
(
y
)
|
2
μ
*
-
2
(
u
n
)
+
(
y
)
ϕ
(
y
)
|
x
-
y
|
μ
𝑑
x
𝑑
y
→
∫
Ω
∫
Ω
|
u
+
(
x
)
|
2
μ
*
|
u
+
(
y
)
|
2
μ
*
-
2
u
+
(
y
)
ϕ
(
y
)
|
x
-
y
|
μ
𝑑
x
𝑑
y
.
Now, for ϕ∈D01,2(Ω) consider
〈
I
′
(
u
n
)
-
I
′
(
u
)
,
ϕ
〉
=
∫
Ω
∇
u
n
.
∇
ϕ
d
x
-
∫
Ω
∫
Ω
|
(
u
n
)
+
(
x
)
|
2
μ
*
|
(
u
n
)
+
(
y
)
|
2
μ
*
-
2
(
u
n
)
+
ϕ
(
y
)
|
x
-
y
|
μ
𝑑
x
𝑑
y
-
∫
Ω
∇
u
.
∇
ϕ
d
x
+
∫
Ω
∫
Ω
|
u
+
(
x
)
|
2
μ
*
|
u
+
(
y
)
|
2
μ
*
-
2
u
+
ϕ
(
y
)
|
x
-
y
|
μ
𝑑
x
𝑑
y
.
By using (4.5) and the fact that un⇀u weakly in D01,2(Ω), the claim follows. By the Brezis–Lieb lemma (see [8, 16]) we have
I
∞
(
v
n
)
=
1
2
∥
u
n
∥
2
-
1
2
∥
u
∥
2
-
1
2
⋅
2
μ
*
∫
Ω
∫
Ω
|
(
u
n
-
u
)
+
(
x
)
|
2
μ
*
|
(
u
n
-
u
)
+
(
y
)
|
2
μ
*
|
x
-
y
|
μ
𝑑
x
𝑑
y
+
o
(
1
)
=
1
2
∥
u
n
∥
2
-
1
2
⋅
2
μ
*
∫
Ω
∫
Ω
|
(
u
n
)
+
(
x
)
|
2
μ
*
|
(
u
n
)
+
(
y
)
|
2
μ
*
|
x
-
y
|
μ
𝑑
x
𝑑
y
-
1
2
∥
u
∥
2
+
1
2
⋅
2
μ
*
∫
Ω
∫
Ω
|
u
+
(
x
)
|
2
μ
*
|
u
+
(
y
)
|
2
μ
*
|
x
-
y
|
μ
𝑑
x
𝑑
y
+
o
(
1
)
=
I
(
u
n
)
-
I
(
u
)
+
o
(
1
)
→
c
-
I
(
u
)
.
Now we will show that I∞′(vn)→0 in (D01,2(Ω))′. By Lemma 4.1, for any ϕ∈D01,2(Ω),
〈
I
∞
′
(
v
n
)
,
ϕ
〉
=
〈
I
′
(
v
n
)
,
ϕ
〉
=
〈
I
′
(
u
n
)
,
ϕ
〉
-
〈
I
′
(
u
)
,
ϕ
〉
+
o
(
1
)
→
0
.
This implies I∞′(vn)→0 in (D01,2(Ω))′. ∎
Lemma 4.3.
Let {yn}⊂Ω and {λn}⊂(0,∞) be such that 1λndist(yn,∂Ω)→∞. Assume the sequence {un} and the rescaled sequence
f
n
(
x
)
=
λ
n
N
-
2
2
u
n
(
λ
n
x
+
y
n
)
is such that fn⇀f weakly in D1,2(RN), fn→f a.e. on RN, I∞(un)→c, I∞′(un)→0 in (D01,2(Ω))′. Then I∞′(f)=0. Also, the sequence
z
n
(
x
)
=
u
n
(
x
)
-
λ
n
2
-
N
2
f
(
x
-
y
n
λ
n
)
satisfies ∥zn∥2=∥un∥2-∥f∥2+o(1), I∞(zn)→c-I∞(f) and I∞′(zn)→0 in (D01,2(Ω))′.
Proof.
For ϕ∈Cc∞(ℝN) define ϕn(x):=λn2-N2ϕ(x-ynλn). If ϕ∈Cc∞(Bk), then for large n, ϕn∈Cc∞(Ω). It implies
〈
I
∞
′
(
f
n
)
,
ϕ
〉
=
〈
I
∞
′
(
u
n
)
,
ϕ
n
〉
≤
∥
I
∞
′
(
u
n
)
∥
∥
ϕ
n
∥
=
∥
I
∞
′
(
u
n
)
∥
∥
ϕ
∥
→
0
.
Hence, I∞′(fn)→0 as n→∞ in (D01,2(Bk))′ for each k.
Claim.
We have I∞′(f)=0.
Since ϕ∈Cc∞(ℝN), we obtain ϕ∈Cc∞(Bk) for some k. Now, using the fact 1λndist(yn,∂Ω)→∞, I∞′(fn)→0 in (D01,2(Bk))′ and following the steps of Claim of Lemma 4.2, we have 〈I∞′(fn)-I∞′(f),ϕ〉→0, that is, the claim holds. By the Brezis–Lieb lemma (see [8, 16]),
I
∞
(
z
n
)
=
I
∞
(
f
n
-
f
)
=
I
∞
(
u
n
)
-
I
∞
(
f
)
+
o
(
1
)
→
c
-
I
∞
(
f
)
.
As fn⇀f weakly in D1,2(ℝN), we obtain
∥
z
n
∥
2
=
∫
ℝ
N
|
∇
u
n
(
x
)
-
λ
n
-
N
2
∇
f
(
x
-
y
n
λ
n
)
|
2
𝑑
x
=
∥
u
n
∥
2
-
∥
f
∥
2
+
o
(
1
)
.
By Lemma 4.1, for any ϕ∈D01,2(Ω), we have
〈
I
∞
′
(
z
n
)
,
ϕ
〉
=
〈
I
∞
′
(
u
n
)
-
I
∞
′
(
λ
n
2
-
N
2
f
(
.
-
y
n
λ
n
)
)
,
ϕ
〉
+
o
(
1
)
=
〈
I
∞
′
(
u
n
)
,
ϕ
〉
+
o
(
1
)
=
o
(
1
)
.
This implies I∞′(zn)→0 in (D01,2(Ω))′. ∎
Before proving the global compactness lemma for the Choquard equation, we will define the well-known Morrey spaces.
Definition 4.4.
A measurable function u:ℝN→ℝ belongs to Morrey space ℒr,γ(ℝN), with r∈[1,∞) and γ∈[0,N], if and only if
∥
u
∥
ℒ
r
,
γ
(
ℝ
N
)
r
:=
sup
R
>
0
,
x
∈
ℝ
N
R
γ
-
N
∫
B
(
x
,
R
)
|
u
|
r
𝑑
y
<
∞
.
By Hölder’s inequality, we have L2*(ℝN)↪ℒ2,N-2(ℝN).
Lemma 4.5 (Global Compactness Lemma).
Let {un}n∈N⊂D01,2(Ω) be such that I(un)→c, I′(un)→0. Then passing if necessary to a subsequence, there exists a solution v0∈D01,2(Ω) of
(4.6)
-
Δ
u
=
(
∫
Ω
|
u
+
(
y
)
|
2
μ
*
|
x
-
y
|
μ
𝑑
y
)
|
u
+
|
2
μ
*
-
1
in
Ω
and (possibly) k∈N∪{0}, nontrivial solutions {v1,v2,…,vk} of
(4.7)
-
Δ
u
=
(
|
x
|
-
μ
*
|
u
+
|
2
μ
*
)
|
u
+
|
2
μ
*
-
1
in
ℝ
N
with vi∈D1,2(RN) and k sequences {yni}n∈N⊂RN and {λni}n∈N⊂R+, i=1,2,…,k, satisfying
1
λ
n
i
dist
(
y
n
i
,
∂
Ω
)
→
∞
𝑎𝑛𝑑
∥
u
n
-
v
0
-
∑
i
=
1
k
(
λ
n
i
)
2
-
N
2
v
i
(
.
-
y
n
i
λ
n
i
)
∥
→
0
,
n
→
∞
,
(4.8)
∥
u
n
∥
2
→
∑
i
=
0
k
∥
v
i
∥
2
,
n
→
∞
,
I
(
v
0
)
+
∑
i
=
1
k
I
∞
(
v
i
)
=
c
.
Proof.
We divide the proof into several steps.
Step 1. By coercivity of the functional I, we get {un} is a bounded sequence in D01,2(Ω). It implies that there exists a v0∈D01,2(Ω) such that un⇀v0 weakly in D01,2(Ω), un→v0 a.e. on Ω. By Lemma 4.2, I′(v0)=0. Set un1=un-v0. Then
(4.9)
∥
u
n
1
∥
2
=
∥
u
n
∥
2
-
∥
v
0
∥
2
+
o
(
1
)
,
I
∞
(
u
n
1
)
→
c
-
I
(
v
0
)
and
I
∞
′
(
u
n
1
)
→
0
in
(
D
0
1
,
2
(
Ω
)
)
′
.
Moreover, there exists a constant M1>0 such that ∥un1∥<M1 for all n∈ℕ.
Step 2. If
∫
Ω
∫
Ω
|
(
u
n
1
)
+
(
x
)
|
2
μ
*
|
(
u
n
1
)
+
(
y
)
|
2
μ
*
|
x
-
y
|
μ
𝑑
x
𝑑
y
→
0
,
then using the fact that I′(un)→0, it follows that un1→0 in D01,2(Ω) and we are done. If
∫
Ω
∫
Ω
|
(
u
n
1
)
+
(
x
)
|
2
μ
*
|
(
u
n
1
)
+
(
y
)
|
2
μ
*
|
x
-
y
|
μ
𝑑
x
𝑑
y
↛
0
,
then we may assume that
∫
Ω
∫
Ω
|
(
u
n
1
)
+
(
x
)
|
2
μ
*
|
(
u
n
1
)
+
(
y
)
|
2
μ
*
|
x
-
y
|
μ
𝑑
x
𝑑
y
>
δ
for some
δ
>
0
.
This together with the Hardy–Littlewood–Sobolev inequality gives ∥un1∥L2*>δ1 for all n and for an appropriate constant δ1>0. Taking into account that un1 is a bounded sequence in L2*(ℝN), L2*(ℝN)↪ℒ2,N-2(ℝN), and [28, Theorem 2], we obtain
c
2
<
∥
u
n
1
∥
ℒ
2
,
N
-
2
(
ℝ
N
)
<
c
1
for all
n
.
Thus, there exists a positive constant C0 such that for all n, we have
(4.10)
C
0
<
∥
u
n
1
∥
ℒ
2
,
N
-
2
(
ℝ
N
)
<
C
0
-
1
.
Now employing the definition of Morrey spaces and (4.10), for each n∈ℕ there exists {yn1,λn1}∈ℝN×ℝ+ such that
0
<
C
0
^
<
∥
u
n
1
∥
ℒ
r
,
γ
(
ℝ
N
)
2
-
C
0
2
2
n
<
(
λ
n
1
)
-
2
∫
B
(
y
n
1
,
λ
n
1
)
|
u
n
1
|
2
𝑑
y
for some suitable positive constant C0^. Now, define
f
n
1
(
x
)
:=
(
λ
n
1
)
N
-
2
2
u
n
1
(
λ
n
1
x
+
y
n
1
)
.
Since ∥fn1∥=∥un1∥, we have ∥fn1∥<M1 for all n∈ℕ and we can assume that fn1⇀v1 weakly in D1,2(ℝN) and fn1→v1 a.e. on ℝN. Moreover,
∫
B
(
0
,
1
)
|
f
n
1
|
2
𝑑
x
=
(
λ
n
1
)
N
-
2
∫
B
(
0
,
1
)
|
u
n
1
(
λ
n
1
x
+
y
n
1
)
|
2
𝑑
x
=
(
λ
n
1
)
-
2
∫
B
(
y
n
1
,
λ
n
1
)
|
u
n
1
(
y
)
|
2
𝑑
y
>
C
0
^
>
0
.
Since, D1,2(ℝN)↪Lloc2(ℝN) is compact, we have ∫B(0,1)|v1|2𝑑x>C0^>0. It implies that v1≠0.
Step 3. We claim that λn→0 and yn1→y0∈Ω¯. Let if possible λn→∞. As {un1} is a bounded sequence in D01,2(Ω), it implies {un1} is a bounded sequence in L2(Ω). Thus, if we define Ωn=Ω-yn1λn1, then
∫
Ω
n
|
f
n
1
|
2
𝑑
x
=
1
(
λ
n
1
)
2
∫
Ω
|
u
n
1
|
2
𝑑
x
≤
C
λ
n
2
→
0
.
Contrary to this, using Fatou’s lemma, we have
0
=
lim inf
n
→
∞
∫
Ω
n
|
f
n
1
|
2
𝑑
x
≥
∫
Ω
n
|
v
1
|
2
𝑑
x
.
This means that v≡0, which is not possible by Step 2. Hence {λn1} is bounded in ℝ, that is, there exists 0≤λ01∈ℝ such that λn1→λ01 as n→∞. If |yn1|→∞, then for any x∈Ω and large n, λnx+yn∉Ω¯. Since un∈D01,2(Ω), it follows that un1(λnx+yn)=0 for all x∈Ω, which yields a contradiction to the assumption
∥
u
n
∥
N
L
2
⋅
2
μ
*
>
δ
>
0
.
Therefore, yn1 is bounded, it implies that yn1→y01∈ℝN. Now let if possible λn1→λ01>0. Then
Ω
n
→
Ω
-
y
0
1
λ
0
1
=
Ω
0
≠
ℝ
N
.
Hence using the fact that un1⇀0 weakly in D01,2(Ω), we have fn1⇀0 weakly in D1,2(ℝN) which is not possible since by Step 2, v1≠0. This implies λn1→0. Arguing by contradiction, we assume that
(4.11)
y
0
1
∉
Ω
¯
.
In view of the fact that λn1x+yn1→y01 for all x∈Ω as n→∞. Now using (4.11), we have λn1x+yn1∉Ω¯ for all x∈Ω and n large enough. It implies that un1(λn1x+yn1)=0 for n large enough, which is not possible. Therefore, y01∈Ω¯. This completes the proof of claim and Step 3.
Step 4: Assume that
lim
n
→
∞
1
λ
n
1
dist
(
y
n
1
,
∂
Ω
)
→
α
<
∞
.
Then v1 is a solution of (2.12) and by Theorem 2.8 we have v1≡0, which is not possible. Therefore,
1
λ
n
1
dist
(
y
n
1
,
∂
Ω
)
→
∞
as
n
→
∞
.
Thus by (4.9) and Lemma 4.3, we have I∞′(v1)=0 and the sequence
u
n
2
(
x
)
=
u
n
1
(
x
)
-
λ
n
2
-
N
2
v
1
(
x
-
y
n
λ
n
)
satisfies
I
∞
(
u
n
2
)
→
c
-
I
∞
(
v
0
)
-
I
∞
(
v
1
)
and
I
∞
′
(
u
n
2
)
→
0
in
(
D
0
1
,
2
(
Ω
)
)
′
.
By Proposition 2.6, we have I∞(v1)≥β. So, iterating the above procedure, we can construct sequences {vi},{λni},{fni} and after k iterations we obtain
I
∞
(
u
n
k
+
1
)
<
I
(
u
n
)
-
I
(
v
0
)
-
∑
i
=
1
k
I
∞
(
v
i
)
≤
I
(
u
n
)
-
I
(
v
0
)
-
k
β
.
As the later will be negative for large k, the induction process terminates after some index k≥0. Consequently, we get k sequences {yni}n⊂Ω and {λni}n⊂ℝ+, satisfying (4.8). ∎
Definition 4.6.
We say that I satisfies the Palais–Smale condition at c if for any sequence uk∈D01,2(Ω) such that I(uk)→c and I′(uk)→0 there exists a subsequence that converges strongly in D01,2(Ω).
Lemma 4.7.
The functional I satisfies the Palais–Smale condition for any c∈(β,2β), where
β
=
1
2
(
N
-
μ
+
2
2
N
-
μ
)
S
H
,
L
2
N
-
μ
N
-
μ
+
2
.
Proof.
For some c∈(β,2β), we assume that there exists {un}∈D01,2(Ω) such that
I
(
u
n
)
→
c
,
I
′
(
u
n
)
→
0
in
(
D
0
1
,
2
(
Ω
)
)
′
.
By Lemma 4.5, passing to a subsequence (if necessary), there exists a solution v0∈D01,2(Ω) of (4.6) and k∈ℕ∪{0}, nontrivial solutions {v1,v2,…,vk} of (4.7) with vi∈D1,2(ℝN) and k sequences {yni}n⊂ℝN and {λni}n⊂ℝ+ satisfying (4.8). Now, by equation (4.8) and Proposition 2.6 we have kβ≤c<2β. This implies k≤1.
If k=0, compactness holds and we are done. If k=1, then we have two possibilities: either v0≢0 or v0≡0. If v0≢0, since I(v0)≥β and by [16, Lemma 1.3], β is never achieved on bounded domain, we have I(v0)>β and this is not possible. If v0≡0, then by Theorem 2.8, I∞(v1)=c and v1 is a nonnegative solution of (4.7).
Next, by Corollary 3.5, we deduce that v1 is radially symmetric, monotonically deceasing and of the form v1(x)=(ab+|x-x0|2)N-22, for some constants a,b>0 and some x0∈ℝN. Therefore by Lemma 2.3, we conclude that SH,L is achieved by v1. It follows that I∞(v1)=β, which is a contradiction since I∞(v1)=c>β. ∎
5 Proof of Theorem 1.1
To prove Theorem 1.1, we shall first establish some auxiliary results.
Let R1,R2 be the radii of the annulus as in Theorem 1.1. Without loss of generality, we can assume x0=0, R1=14R, R2=4R, where R>0 will be chosen sufficiently large. Consider the family of functions
u
t
σ
(
x
)
:=
S
(
N
-
μ
)
(
2
-
N
)
4
(
N
-
μ
+
2
)
C
(
N
,
μ
)
2
-
N
2
(
N
-
μ
+
2
)
(
1
-
t
(
1
-
t
)
2
+
|
x
-
t
σ
|
2
)
N
-
2
2
∈
D
1
,
2
(
ℝ
N
)
,
where σ∈Σ:={x∈ℝN:|x|=1},t∈[0,1). Note that if t→1, then utσ concentrates at σ. Also, if t→0 then
u
t
σ
→
u
0
:=
S
(
N
-
μ
)
(
2
-
N
)
4
(
N
-
μ
+
2
)
C
(
N
,
μ
)
2
-
N
2
(
N
-
μ
+
2
)
(
1
1
+
|
x
|
2
)
N
-
2
2
.
Now, define υ∈Cc∞(Ω) such that 0≤υ≤1 on Ω and
υ
(
x
)
=
{
1
,
1
2
<
|
x
|
<
2
,
0
,
|
x
|
>
4
,
|
x
|
<
1
4
.
Subsequently, we can define
υ
R
(
x
)
=
{
υ
(
R
x
)
,
0
<
|
x
|
<
1
2
R
,
1
,
1
2
R
≤
|
x
|
≤
R
,
υ
(
x
R
)
,
|
x
|
≥
R
.
We now define
g
t
σ
(
x
)
=
u
t
σ
(
x
)
υ
R
(
x
)
∈
D
0
1
,
2
(
Ω
)
,
g
0
(
x
)
=
u
0
(
x
)
υ
R
(
x
)
.
We establish the following auxiliary result.
Lemma 5.1.
Let σ∈Σ and t∈(0,1]. Then the following holds:
∥
(
u
t
σ
)
+
∥
N
L
=
∥
(
u
0
)
+
∥
N
L
,
∥
u
t
σ
∥
2
=
S
H
,
L
∥
(
u
t
σ
)
+
∥
N
L
2
,
lim
R
→
∞
sup
σ
∈
Σ
,
t
∈
[
0
,
1
)
∥
g
t
σ
-
u
t
σ
∥
=
0
,
lim
R
→
∞
sup
σ
∈
Σ
,
t
∈
[
0
,
1
)
∥
g
t
σ
∥
N
L
2
⋅
2
μ
*
=
∥
u
t
σ
∥
N
L
2
⋅
2
μ
*
.
Proof.
By trivial transformations, we can get first two properties utσ and since utσ is a minimizer of SH,L therefore, third ones holds.
We have
(5.1)
∫
ℝ
N
|
∇
g
t
σ
-
∇
u
t
σ
|
2
𝑑
x
≤
2
∫
ℝ
N
|
u
t
σ
(
x
)
∇
υ
R
(
x
)
|
2
𝑑
x
+
2
∫
ℝ
N
|
∇
u
t
σ
(
x
)
υ
R
(
x
)
-
∇
u
t
σ
(
x
)
|
2
𝑑
x
≤
C
(
R
2
∫
B
1
2
R
|
u
t
σ
(
x
)
|
2
𝑑
x
+
∫
B
1
2
R
|
∇
u
t
σ
(
x
)
|
2
𝑑
x
)
+
C
(
1
R
2
∫
B
4
R
∖
B
2
R
|
u
t
σ
(
x
)
|
2
𝑑
x
+
∫
ℝ
N
∖
B
2
R
|
∇
u
t
σ
(
x
)
|
2
𝑑
x
)
,
where Bα is a ball of radius α and center 0.
From the definition of utσ, we have
R
2
∫
B
1
2
R
|
u
t
σ
(
x
)
|
2
𝑑
x
≤
C
R
2
∫
B
1
2
R
𝑑
x
≤
C
R
N
-
2
,
∫
B
1
2
R
|
∇
u
t
σ
(
x
)
|
2
𝑑
x
≤
C
∫
B
1
2
R
|
x
-
t
σ
|
𝑑
x
≤
C
∫
B
1
2
R
𝑑
x
≤
C
R
N
,
and
1
R
2
∫
B
4
R
∖
B
2
R
|
u
t
σ
(
x
)
|
2
𝑑
x
≤
C
R
2
∫
B
4
R
∖
B
2
R
1
|
x
|
2
N
-
4
𝑑
x
≤
C
R
N
-
2
,
∫
ℝ
N
∖
B
2
R
|
∇
u
t
σ
(
x
)
|
2
𝑑
x
≤
C
∫
ℝ
N
∖
B
2
R
1
|
x
|
2
N
-
2
𝑑
x
≤
C
R
N
-
2
.
Therefore, from (5.1) if R→∞, we get supσ∈Σ,t∈(0,1]∥gtσ-utσ∥→0.
Next, we shall prove that
lim
R
→
∞
sup
σ
∈
Σ
,
t
∈
(
0
,
1
]
∥
g
t
σ
∥
N
L
2
⋅
2
μ
*
=
∥
u
t
σ
∥
N
L
2
⋅
2
μ
*
.
Consider
∥
g
t
σ
∥
N
L
2
⋅
2
μ
*
-
∥
u
t
σ
∥
N
L
2
⋅
2
μ
*
=
∫
ℝ
N
∫
ℝ
N
(
υ
R
2
μ
*
(
x
)
υ
R
2
μ
*
(
y
)
-
1
)
|
u
t
σ
(
x
)
|
2
μ
*
|
u
t
σ
(
y
)
|
2
μ
*
|
x
-
y
|
μ
𝑑
x
𝑑
y
≤
C
∑
i
=
1
5
J
i
,
where
J
1
=
∫
B
2
R
∖
B
1
2
R
∫
B
1
2
R
|
u
t
σ
(
x
)
|
2
μ
*
|
u
t
σ
(
y
)
|
2
μ
*
|
x
-
y
|
μ
𝑑
x
𝑑
y
,
J
2
=
∫
B
2
R
∖
B
1
2
R
∫
ℝ
N
∖
B
2
R
|
u
t
σ
(
x
)
|
2
μ
*
|
u
t
σ
(
y
)
|
2
μ
*
|
x
-
y
|
μ
𝑑
x
𝑑
y
,
J
3
=
∫
B
1
2
R
∫
B
1
2
R
|
u
t
σ
(
x
)
|
2
μ
*
|
u
t
σ
(
y
)
|
2
μ
*
|
x
-
y
|
μ
𝑑
x
𝑑
y
,
J
4
=
∫
B
1
2
R
∫
ℝ
N
∖
B
2
R
|
u
t
σ
(
x
)
|
2
μ
*
|
u
t
σ
(
y
)
|
2
μ
*
|
x
-
y
|
μ
𝑑
x
𝑑
y
,
J
5
=
∫
ℝ
N
∖
B
2
R
∫
ℝ
N
∖
B
2
R
|
u
t
σ
(
x
)
|
2
μ
*
|
u
t
σ
(
y
)
|
2
μ
*
|
x
-
y
|
μ
𝑑
x
𝑑
y
.
By the Hardy–Littlewood–Sobolev inequality, we have the following estimates:
J
1
≤
C
(
N
,
μ
)
(
∫
B
1
2
R
(
1
-
t
)
N
d
x
(
(
1
-
t
)
2
+
|
x
-
t
σ
|
2
)
N
)
2
N
-
μ
2
N
(
∫
B
2
R
∖
B
1
2
R
(
1
-
t
)
N
d
x
(
(
1
-
t
)
2
+
|
x
-
t
σ
|
2
)
N
)
2
N
-
μ
2
N
≤
C
(
∫
B
1
2
R
(
1
-
t
)
N
-
2
𝑑
x
)
2
N
-
μ
2
N
≤
C
(
1
2
R
)
2
N
-
μ
2
,
J
2
≤
C
(
N
,
μ
)
(
∫
B
2
R
∖
B
1
2
R
(
1
-
t
)
N
d
x
(
(
1
-
t
)
2
+
|
x
-
t
σ
|
2
)
N
)
2
N
-
μ
2
N
(
∫
ℝ
N
∖
B
2
R
(
1
-
t
)
N
d
x
(
(
1
-
t
)
2
+
|
x
-
t
σ
|
2
)
N
)
2
N
-
μ
2
N
≤
C
(
∫
ℝ
N
∖
B
2
R
d
x
|
x
-
t
σ
|
2
N
)
2
N
-
μ
2
N
≤
C
(
∫
|
y
+
t
σ
|
≥
2
R
d
y
|
y
|
2
N
)
2
N
-
μ
2
N
≤
C
(
∫
|
y
|
≥
2
R
-
1
d
y
|
y
|
2
N
)
2
N
-
μ
2
N
≤
C
(
1
2
R
-
1
)
2
N
-
μ
2
,
J
3
≤
C
(
N
,
μ
)
(
∫
B
1
2
R
(
1
-
t
)
N
d
x
(
(
1
-
t
)
2
+
|
x
-
t
σ
|
2
)
N
)
2
N
-
μ
N
≤
C
(
∫
B
1
2
R
(
1
-
t
)
N
-
2
𝑑
x
)
2
N
-
μ
N
≤
C
(
1
2
R
)
2
N
-
μ
.
Using the same estimates as above, we can easily obtain
J
4
≤
C
(
1
2
R
)
2
N
-
μ
2
and
J
5
≤
C
(
1
2
R
-
1
)
2
N
-
μ
.
This implies that
sup
σ
∈
Σ
,
t
∈
[
0
,
1
)
(
∥
g
t
σ
∥
N
L
2
⋅
2
μ
*
-
∥
u
t
σ
∥
N
L
2
⋅
2
μ
*
)
→
0
as
R
→
∞
and completes the proof. ∎
In order to proceed further we define the manifold ℳ and the functions G:ℳ→ℝN as follows:
ℳ
=
{
u
∈
D
0
1
,
2
(
Ω
)
:
∫
Ω
∫
Ω
|
u
+
(
x
)
|
2
μ
*
|
u
+
(
y
)
|
2
μ
*
|
x
-
y
|
μ
𝑑
x
𝑑
y
=
1
}
and
G
(
u
)
=
∫
Ω
x
|
∇
u
|
2
𝑑
x
.
We also define SH,L(u,Ω):D01,2(Ω)∖{0}→ℝ, SH,L:D1,2(ℝN)∖{0}→ℝ and τ:D01,2(Ω)→ℝ as
S
H
,
L
(
u
,
Ω
)
=
∫
Ω
|
∇
u
|
2
𝑑
x
(
∫
Ω
∫
Ω
|
u
+
(
x
)
|
2
μ
*
|
u
+
(
y
)
|
2
μ
*
|
x
-
y
|
μ
𝑑
x
𝑑
y
)
1
2
μ
*
,
S
H
,
L
(
u
)
=
∫
ℝ
N
|
∇
u
|
2
𝑑
x
∥
u
+
∥
N
L
2
,
τ
(
u
)
=
(
∫
Ω
∫
Ω
|
u
+
(
x
)
|
2
μ
*
|
u
+
(
y
)
|
2
μ
*
|
x
-
y
|
μ
𝑑
x
𝑑
y
)
1
2
μ
*
.
Proposition 5.2.
If SH,L(⋅,Ω)∈C1(D01,2(Ω)∖{0}) and SH,L′(u,Ω)=0 for u∈D01,2(Ω), then one has I′(λu)=0 for some λ>0.
Proof.
Let w∈D01,2(Ω). Then
〈
S
H
,
L
′
(
u
,
Ω
)
,
w
〉
=
2
τ
(
u
)
∫
Ω
∇
u
.
∇
w
d
x
-
2
∥
u
∥
2
τ
(
u
)
1
-
2
μ
*
∫
Ω
∫
Ω
|
u
+
(
x
)
|
2
μ
*
|
u
+
(
y
)
|
2
μ
*
-
2
u
+
(
y
)
w
(
y
)
|
x
-
y
|
μ
𝑑
x
𝑑
y
τ
(
u
)
2
.
As SH,L′(u,Ω)(w)=0, it implies
τ
(
u
)
∫
Ω
∇
u
.
∇
w
d
x
=
∥
u
∥
2
τ
(
u
)
1
-
2
μ
*
∫
Ω
∫
Ω
|
u
+
(
x
)
|
2
μ
*
|
u
+
(
y
)
|
2
μ
*
-
2
u
+
(
y
)
w
(
y
)
|
x
-
y
|
μ
𝑑
x
𝑑
y
,
that is,
∫
Ω
∇
u
.
∇
w
d
x
=
∥
u
∥
2
∫
Ω
∫
Ω
|
u
+
(
x
)
|
2
μ
*
|
u
+
(
y
)
|
2
μ
*
-
2
u
+
(
y
)
w
(
y
)
|
x
-
y
|
μ
𝑑
x
𝑑
y
∫
Ω
∫
Ω
|
u
+
(
x
)
|
2
μ
*
|
u
+
(
y
)
|
2
μ
*
|
x
-
y
|
μ
𝑑
x
𝑑
y
.
Therefore, if we choose
λ
2
(
2
μ
*
-
1
)
=
∥
u
∥
2
∫
Ω
∫
Ω
|
u
+
(
x
)
|
2
μ
*
|
u
+
(
y
)
|
2
μ
*
|
x
-
y
|
μ
𝑑
x
𝑑
y
,
then we get I′(λu)=0. ∎
Proposition 5.3.
Let {vn}⊂M be a Palais–Smale sequence for SH,L(⋅,Ω) at level c. Then the sequence {un} given by
u
n
=
λ
n
v
n
,
λ
n
=
(
S
H
,
L
(
v
n
,
Ω
)
)
N
-
2
2
(
N
-
μ
+
2
)
is a Palais–Smale sequence for I at level N-μ+22(2N-μ)c2N-μN-μ+2.
Proof.
By the calculations of Proposition 5.2 for any w∈D01,2(Ω), we have
1
2
〈
S
H
,
L
′
(
v
n
,
Ω
)
,
w
〉
=
∫
Ω
∇
v
n
.
∇
w
d
x
-
λ
n
2
(
2
μ
*
-
1
)
∫
Ω
∫
Ω
|
(
v
n
)
+
(
x
)
|
2
μ
*
|
(
v
n
)
+
(
y
)
|
2
μ
*
-
2
(
v
n
)
+
(
y
)
w
(
y
)
|
x
-
y
|
μ
𝑑
x
𝑑
y
.
Now by multiplying the above equation by λn for any w∈D01,2(Ω), we obtain
〈
I
′
(
u
n
)
,
w
〉
=
∫
Ω
∇
u
n
.
∇
w
d
x
-
∫
Ω
∫
Ω
|
(
u
n
)
+
(
x
)
|
2
μ
*
|
(
u
n
)
+
(
y
)
|
2
μ
*
-
2
(
u
n
)
+
(
y
)
w
(
y
)
|
x
-
y
|
μ
𝑑
x
𝑑
y
.
Since vn∈ℳ, it follows that λ2(2μ*-1)=∥vn∥2=SH,L(vn,Ω), that is,
λ
n
=
S
H
,
L
(
v
n
,
Ω
)
N
-
2
2
(
N
-
μ
+
2
)
.
From SH,L(vn,Ω)=c+o(1) we get λn is bounded. In particular, it follows that 〈I′(λnvn), w〉→0 as n→∞. Also, we have un is bounded and this yields
o
(
1
)
=
〈
I
′
(
u
n
)
,
u
n
〉
=
∥
u
n
∥
2
-
∫
Ω
∫
Ω
|
(
u
n
)
+
(
x
)
|
2
μ
*
|
(
u
n
)
+
(
y
)
|
2
μ
*
|
x
-
y
|
μ
𝑑
x
𝑑
y
.
All the above facts imply that
lim
n
→
∞
I
(
u
n
)
=
N
-
μ
+
2
2
(
2
N
-
μ
)
lim
n
→
∞
λ
n
2
⋅
2
μ
*
=
N
-
μ
+
2
2
(
2
N
-
μ
)
c
2
N
-
μ
N
-
μ
+
2
.
∎
Remark 5.4.
Since we proved that I satisfies the Palais–Smale condition in (β,2β), it follows that SH,L(⋅,Ω) satisfies the Palais–Smale condition in (SH,L,2N-μ+22N-μSH,L) by using Proposition 5.2.
Lemma 5.5.
If ftσ(x):=gtσ(x)∥gtσ∥NL and f0(x):=g0(x)∥g0∥NL, then
lim
R
→
∞
S
H
,
L
(
f
t
σ
,
Ω
)
=
S
H
,
L
(
u
t
σ
)
=
S
H
,
L
uniformly with respect to σ∈Σ and t∈[0,1).
Proof.
This is a trivial consequence of Lemma 5.1. ∎
In particular, if R>1 sufficiently large, then we can achieve that
sup
σ
,
t
(
f
t
σ
,
Ω
)
<
S
1
<
2
N
-
μ
+
2
2
N
-
μ
S
H
,
L
for some
S
1
∈
ℝ
.
Proof of Theorem 1.1 completed.
As we have established, SH,L(⋅,Ω) satisfies Palais–Smale at level α on ℳ for α∈(SH,L,2N-μ+22N-μSH,L). We will argue by contradiction. If SH,L(⋅,Ω) does not admit a critical value in this range, by the deformation lemma (see Bonnet [7, Theorem 2.5]) for any α∈(SH,L,2N-μ+22N-μSH,L) there exist δ>0 and an onto homeomorphism function ψ:ℳ→ℳ such that
ψ
(
ℳ
α
+
δ
)
⊂
ℳ
α
-
δ
,
where ℳα={u∈ℳ:SH,L(u,Ω)<α}. For a given fixed ε>0 we can cover the interval [SH,L+ε,S1] by finitely many such δ-intervals and composing the deformation maps, we get an onto-homeomorphism function ψ:ℳ→ℳ such that
ψ
(
ℳ
S
1
)
⊂
ℳ
S
H
,
L
+
ε
.
Also, we can assume ψ(u)=u for all u whenever SH,L(u,Ω)≤SH,L+ε2.
By the concentration-compactness lemma (see [14]) and [16, Lemma 1.2], we have that for any sequence {um}∈ℳSH,L+1m there exists a subsequence and x(0)∈Ω¯ such that
(
∫
Ω
|
(
u
m
)
+
(
y
)
|
2
μ
*
|
x
-
y
|
μ
𝑑
y
)
|
(
u
m
)
+
|
2
μ
*
d
x
⇁
δ
x
(
0
)
,
|
∇
u
m
|
2
d
x
⇁
S
H
,
L
δ
x
(
0
)
weakly in the sense of measure. This implies given any neighborhood V of Ω¯, there exists an ε>0 such that G(ℳSH,L)⊂V.
Since Ω is a smooth bounded domain, we can find a neighborhood V of Ω¯ such that for any q∈V there exists a unique nearest neighbor r=π(q)∈Ω¯ such that the projection π is continuous. Let ε be chosen for such a neighborhood V, and let ψ:ℳ→ℳ be the corresponding onto homeomorphism. Define the map D:Σ×[0,1]→Ω¯ given by
D
(
σ
,
t
)
=
π
(
G
(
ψ
(
f
t
σ
)
)
)
.
It is easy to see that D is well-defined, continuous and satisfies
D
(
σ
,
0
)
=
π
(
G
(
ψ
(
f
0
)
)
)
=
:
y
0
∈
Ω
¯
and
D
(
σ
,
1
)
=
σ
for all
σ
∈
Σ
.
This implies that D is a contraction of Σ in Ω¯ contradicting the hypothesis of Ω. Hence, our assumption is wrong implies that SH,L(⋅,Ω) has a critical value, that is, there exists a u∈D01,2(Ω) such that u is a solution to problem (P). Now, using [15, Lemma 4.4], we have u∈L∞(Ω)∩C2(Ω¯). Thus, by the maximum principle, u is a positive solution of problem (P). Hence the proof of Theorem 1.1 is complete. ∎
and
K. Sreenadh
