1 Introduction
The Choquard system
(1.1)
{
-
Δ
u
+
V
(
x
)
u
+
W
u
(
x
,
u
)
=
λ
ϕ
(
x
)
u
,
x
∈
ℝ
3
,
-
Δ
ϕ
=
u
2
,
x
∈
ℝ
3
,
has been studied extensively by many researchers, where V:ℝ3→ℝ, λ is a positive constant, and Ws(x,s):=∂∂sW(x,s), with W:ℝ3×ℝ→ℝ.
Choquard systems such as (1.1) arise in quantum mechanics, and are related to the study of the nonlinear Schrödinger equation for a particle in an electromagnetic field or the Hartree–Fock equation. Such systems have attracted much attention in recent years. For example, Catto, Bris and Lions [14, 4] used the system
{
-
Δ
u
+
ω
u
+
|
u
|
4
/
3
u
=
ϕ
(
x
)
u
,
x
∈
ℝ
3
,
-
Δ
ϕ
=
u
2
,
x
∈
ℝ
3
,
to describe a Hartree model for crystals, where ω is a positive constant. Problem (1.1) can be seen as a nonlinear perturbation of the Choquard system
{
-
Δ
u
+
ω
u
=
ϕ
(
x
)
u
,
x
∈
ℝ
3
,
-
Δ
ϕ
=
u
2
,
x
∈
ℝ
3
.
This system was introduced as an approximation to the Hartree–Fock theory for a one-component plasma [17, 19, 20, 21, 22]. System (1.1) is also related to the so-called Schrödinger–Poisson system
(1.2)
{
-
Δ
u
+
V
(
x
)
u
+
λ
ϕ
(
x
)
u
=
f
(
x
,
u
)
,
x
∈
ℝ
3
,
-
Δ
ϕ
=
u
2
,
x
∈
ℝ
3
.
In the past decade, there have been numerous studies of (1.2) under various assumptions on V(x) and f(x,u), for example, see [1, 2, 3, 7, 8, 11, 13, 15, 24, 25] and the references therein.
Recently, Chen and Xiao [5] studied the equation
(1.3)
{
-
Δ
u
+
V
(
x
)
u
+
|
u
|
p
-
2
u
=
λ
ϕ
u
,
x
∈
ℝ
3
,
-
Δ
ϕ
=
u
2
,
x
∈
ℝ
3
,
where the potential V(x) satisfies:
V
(
x
)
∈
C
(
ℝ
3
)
is 1-periodic in xj for j=1,2,3 and 0 is in a spectral gap (-α,β) of -Δ+V, where 0<α,β<+∞.
Under condition (V), assume 3<p<6. By using a penalization technique and the infinite-dimensional linking theorem, Chen and Xiao [5] proved that there is a nontrivial solution for problem (1.3) for λ>0 small enough.
In [23], Mugnai studied the following system:
(1.4)
{
-
Δ
u
+
ω
u
+
W
u
(
x
,
u
)
=
λ
ϕ
(
x
)
u
,
x
∈
ℝ
3
,
-
Δ
ϕ
=
u
2
,
x
∈
ℝ
3
,
where ω, λ are positive constants, Ws(x,s):=∂∂sW(x,s), and the potential W(x,s) satisfies the following assumptions:
The potential W:ℝ3×ℝ→[0,∞) is such that the derivative Ws:ℝ3×ℝ→ℝ is a Carathéodory function, W(x,s)=W(|x|,s) for a.e. x∈ℝ3 and for every s∈ℝ, and W(x,0)=Ws(x,0)=0 for a.e. x∈ℝ3.
There exist C1,C2>0 and 1<q<p<5 such that |Ws(x,s)|≤C1|s|q+C2|s|p for every s∈ℝ and a.e. x∈ℝ3.
There exists k≥2 such that 0≤sWs(x,s)≤kW(x,s) for every s∈ℝ and a.e. x∈ℝ3.
Under assumptions (W1)–(W3), Mugnai obtained the existence and nonexistence of solutions for system (1.4). In particular, if W(x,s) satisfies (W1), (W2) with p∈(1,3), and (W3) with k≤4, then for any λ,ω>0, there exist nontrivial radial functions (u,ϕ)∈H1(ℝ3)×𝒟1,2(ℝ3) which solve (1.4).
Based on the above facts, a natural question is whether system (1.4) has positive ground state solutions. The present paper is devoted to this aspect and gives an affirmative answer.
We first consider the following autonomous Choquard system:
(1.5)
{
-
Δ
u
+
ω
u
+
W
u
(
u
)
=
ϕ
(
x
)
u
,
x
∈
ℝ
3
,
-
Δ
ϕ
=
u
2
,
x
∈
ℝ
3
,
where ω>0 is a parameter and Ws(s):=ddsW(s), satisfying the following hypotheses:
W
(
s
)
∈
𝒞
1
(
ℝ
,
ℝ
)
is nonnegative and can be written as W(s):=W1(s)-W2(s), where W1(s), W2(s) are nondecreasing functions on [0,+∞).
There exist C1, C2>0 and 2<q<p≤6 such that |W1(s)|+|W2(s)|≤C1|s|q+C2|s|p for every s∈ℝ.
As we are interested in positive solutions, we define W(s)=0 for s<0. We remark that the function W(s):=∑i=1kci|s|pi satisfies (W4)–(W5) if ci>0 and 2≤pi≤6. Obviously, W(s) can be of critical growth, which is different to subcritical growth in the previous works.
Our main results of this paper are as follows.
Theorem 1.1.
Assume (W4)–(W5). Then, for each ω>0, system (1.5) admits a positive ground state solution (u,ϕu).
Our next result is about the properties of the positive ground state solutions for system (1.5).
Theorem 1.2.
If the assumptions of Theorem 1.1 hold, then each positive ground state solution of problem (1.5) is radially symmetric about some point and decays exponentially at infinity.
Next, we consider the following nonautonomous Choquard system:
(1.6)
{
-
Δ
u
+
V
(
x
)
u
+
W
u
(
u
)
=
ϕ
(
x
)
u
,
x
∈
ℝ
3
,
-
Δ
ϕ
=
u
2
,
x
∈
ℝ
3
,
where the potential function V∈𝒞1(ℝ3,ℝ) satisfies
V
(
x
)
≤
lim
inf
|
y
|
→
+
∞
V
(
y
)
=
V
∞
<
+
∞
and the inequality is strict in a subset of positive Lebesgue measure,
2
V
(
x
)
-
〈
∇
V
(
x
)
,
x
〉
≥
C
>
0
for all x∈ℝ3,
there exists a constant C0>0 such that
C
0
=
inf
u
∈
H
1
(
ℝ
N
)
∖
{
0
}
∫
ℝ
3
|
∇
u
|
2
+
V
(
x
)
u
2
d
x
∫
ℝ
3
u
2
𝑑
x
.
Our result on system (1.6) can be stated as follows.
Theorem 1.3.
Assume (W4)–(W5) and (V1)–(V3). Then system (1.6) admits a positive ground state solution.
Remark 1.1.
We remark that, in Theorems 1.1–1.3, we do not assume any (AR) type conditions. In the case Wu(|x|,u)=Wu(u)=|u|p-1u, Mugnai [23] proved that if p∈(1,3), then for any ω, λ>0, there exist nontrivial radial functions (u,ϕu)∈H1(ℝ3)×𝒟1,2(ℝ3) which solve (1.4), see [23, Theorem 1.2]. However, in this case, by our Theorem 1.1, we actually know that system (1.4) admits a positive ground state solution for all p∈[1,5].
Now, we briefly introduce our main idea for proving our main results. Because of the competitive interplay of the nonlinearities ∫ℝ3W(u)𝑑x and ∫ℝ3ϕuu2𝑑x, the classical Nehari manifold method cannot be applied for system (1.5). Inspired by [16], we use the so-called Pohozeav manifold method, but we make some crucial modifications. Unlike [16], we prove directly that the minimizers of cω are solutions of system (1.5), i.e., we do not need to prove the set 𝒫 is a manifold. To prove that cω is achieved, we use symmetric-decreasing rearrangement arguments, similar arguments have been used in [10] for the Kirchhoff problem. The symmetric-decreasing rearrangement arguments are also used in the proof of Theorem 1.2.
Since the potential V(x) is only assumed bounded, the energy functional IV corresponding to system (1.6) does not satisfy the Palais–Smale condition. To overcome this difficulty, we first use the monotonicity technique which was developed by Jeanjean [12]. Then, via the global compactness lemma, we complete the proof of Theorem 1.3.
The paper is organized as follows. In Section 2, we study the autonomous case and prove Theorem 1.1 and Theorem 1.2. In Section 3, by the monotonicity technique and the global compactness lemma, we show that system (1.6) admits a positive ground state solution.
2 The Autonomous Case
In this section, we use the constrained minimization method to study the existence of a positive ground state solution for the Choquard system (1.5).
Denote by H1(ℝ3) the usual Sobolev space equipped with the inner product and norm
(
u
,
v
)
ω
:=
∫
ℝ
3
∇
u
∇
v
+
ω
u
v
d
x
,
∥
u
∥
ω
=
(
u
,
u
)
1
/
2
.
For u∈H1(ℝ3), let ϕu be the unique solution of -Δϕ=u2 in 𝒟1,2(ℝ3). Then
ϕ
u
(
x
)
=
1
4
π
∫
ℝ
3
u
2
(
y
)
|
x
-
y
|
𝑑
y
.
Define the energy functional Iω:H1(ℝ3)→ℝ by
I
ω
(
u
)
:=
1
2
∫
ℝ
3
(
|
∇
u
|
2
+
ω
u
2
)
𝑑
x
+
∫
ℝ
3
W
(
u
)
𝑑
x
-
1
4
∫
ℝ
3
ϕ
u
u
2
𝑑
x
.
Obviously, the functional Iω belongs to 𝒞1(H1(ℝ3),ℝ). Moreover, for any u,φ∈H1(ℝ3), we have
〈
I
ω
′
(
u
)
,
φ
〉
=
∫
ℝ
3
(
∇
u
∇
φ
+
ω
u
φ
)
𝑑
x
+
∫
ℝ
3
W
u
(
u
)
φ
𝑑
x
-
∫
ℝ
3
ϕ
u
u
φ
𝑑
x
.
Clearly, if u is a critical point of Iω, then the pair (u,ϕu) solves the nonlocal problem (1.5).
By Lemma 2.6 in this section, we know that each weak solution for system (1.5) is a classical solution. Then each nontrivial solution u∈H1(ℝ3) of (1.5) satisfies the following Pohozaev identity (see [6, Lemma 3.1]):
P
ω
(
u
)
:=
1
2
∫
ℝ
3
|
∇
u
|
2
d
x
+
3
2
∫
ℝ
3
ω
u
2
d
x
+
3
∫
ℝ
3
W
(
u
)
d
x
-
5
4
∫
ℝ
3
ϕ
u
u
2
d
x
=
0
.
Next, we define
𝒫
ω
:=
{
u
∈
H
1
(
ℝ
3
)
∖
{
0
}
∣
P
ω
(
u
)
=
0
}
,
and consider the following constrained minimization problem:
c
ω
:=
inf
u
∈
𝒫
ω
I
ω
(
u
)
.
Now, we prove the set 𝒫ω is nonempty.
Lemma 2.1.
The set Pω is nonempty and cω>0. If u∈Pω, then there exists C0>0 such that ∥u∥ω>C0.
Proof.
For any u∈H1(ℝ3), we define ut:=u(xt). We claim that the function Iω(ut):(0,+∞)→ℝ has a unique critical point tu∈(0,+∞).
In fact, ddtIω(ut)=0 is equivalent to
1
2
∫
ℝ
3
|
∇
u
|
2
d
x
+
3
t
2
2
∫
ℝ
3
ω
u
2
d
x
+
3
t
2
∫
ℝ
3
W
(
u
)
d
x
-
5
t
4
4
∫
ℝ
3
ϕ
u
u
2
d
x
=
0
.
Obviously, h(t):=ddtIω(ut)=0 has a unique solution in (0,+∞). This implies that 𝒫ω is nonempty.
Let u∈𝒫ω. Then Pω(u)=0, i.e.,
1
2
∫
ℝ
3
|
∇
u
|
2
d
x
+
3
2
∫
ℝ
3
ω
u
2
d
x
+
3
∫
ℝ
3
W
(
u
)
d
x
-
5
4
∫
ℝ
3
ϕ
u
u
2
d
x
=
0
.
By (W4), the Hardy–Littlewood–Sobolev inequality (see [18, Theorem 4.3]) and the Sobolev embedding theorem, we can deduce that
1
2
∥
u
∥
ω
2
≤
1
2
∫
ℝ
3
|
∇
u
|
2
d
x
+
3
2
∫
ℝ
3
ω
u
2
d
x
+
3
∫
ℝ
3
W
(
u
)
d
x
=
5
4
∫
ℝ
3
ϕ
u
u
2
d
x
≤
C
∥
u
∥
L
12
/
5
5
/
3
≤
C
∥
u
∥
ω
4
.
Thus, there exists C0>0 such that
∥
u
∥
ω
2
≥
C
0
.
Therefore, for u∈𝒫ω, one has
I
ω
(
u
)
=
I
ω
(
u
)
-
1
5
P
ω
(
u
)
=
2
5
∫
ℝ
3
|
∇
u
|
2
d
x
+
1
5
∫
ℝ
3
ω
u
2
d
x
+
2
5
∫
ℝ
3
W
(
u
)
d
x
≥
1
5
C
0
,
which implies that cω>0. ∎
Lemma 2.2.
If u∈H1(R3)∖{0}, then tu∈(0,+∞) obtained by Lemma 2.1 is the unique maximum point of the function H(t):=Iω(ut) in (0,+∞).
Proof.
It follows from Lemma 2.1 that tu is the unique critical point of H(t) in (0,+∞). Since
H
(
t
)
=
t
2
∫
ℝ
3
|
∇
u
|
2
d
x
+
t
3
2
∫
ℝ
3
ω
u
2
d
x
+
t
3
∫
ℝ
3
W
(
u
)
d
x
-
t
5
4
∫
ℝ
3
ϕ
u
u
2
d
x
.
Obviously, H(t)→-∞ as t→+∞, H(0)=0 and H(t) is an increasing function with respect to t for t>0 small enough. Thus, tu is the unique maximum point of the function H(t) in (0,+∞). ∎
Lemma 2.3.
Assume that u∈H1(R3)∖{0} is such that Pω(u)≤0. Then tu∈(0,1], where tu is obtained by Lemma 2.1.
Proof.
For u∈H1(ℝ3), it follows from u(x/tu)∈𝒫ω that
(2.1)
t
u
2
∫
ℝ
3
|
∇
u
|
2
d
x
+
3
t
u
3
2
∫
ℝ
3
ω
u
2
d
x
+
3
t
u
3
∫
ℝ
3
W
(
u
)
d
x
-
5
t
u
5
4
∫
ℝ
3
ϕ
u
u
2
d
x
=
0
.
On the other hand,
(2.2)
P
ω
(
u
)
=
1
2
∫
ℝ
3
|
∇
u
|
2
d
x
+
3
2
∫
ℝ
3
ω
u
2
d
x
+
3
∫
ℝ
3
W
(
u
)
d
x
-
5
4
∫
ℝ
3
ϕ
u
u
2
d
x
≤
0
.
Thus, (2.1) and (2.2) imply that tu∈(0,1]. ∎
Lemma 2.4.
Assume (W4)–(W5). Then cω is achieved.
Proof.
Let {un}∈𝒫ω be a minimizing sequence of cω, and let vn be the symmetric-decreasing rearrangement of un. Then, by [18, (iv)–(v) of Chapter 3.3, Theorem 3.7 and Lemma 7.17], we have
(2.3)
∫
ℝ
3
|
∇
v
n
|
2
d
x
≤
∫
ℝ
3
|
∇
u
n
|
2
d
x
,
∫
ℝ
3
W
(
v
n
)
d
x
=
∫
ℝ
3
W
(
u
n
)
d
x
,
and
∫
ℝ
3
ϕ
u
n
u
n
2
d
x
≤
∫
ℝ
3
ϕ
v
n
v
n
2
d
x
,
∫
ℝ
3
|
v
n
|
q
d
x
=
∫
ℝ
3
|
u
n
|
q
d
x
for
2
≤
q
≤
6
.
Since {un} is bounded in H1(ℝ3), {vn} is bounded in H1(ℝ3). Then there exist v∈H1(ℝ3) such that vn⇀v weakly in Hrad1(ℝ3). By the compactness embedding of Hrad1(ℝ3)↪Lp(ℝ3) for 2<p<6, we have that
∫
ℝ
3
ϕ
v
n
v
n
2
𝑑
x
→
∫
ℝ
3
ϕ
v
v
2
𝑑
x
as
n
→
∞
.
On the other hand,
1
2
C
0
≤
1
2
∥
u
n
∥
ω
2
≤
1
2
∫
ℝ
3
|
∇
u
n
|
2
d
x
+
3
2
∫
ℝ
3
ω
u
n
2
d
x
+
3
∫
ℝ
3
W
(
u
n
)
d
x
=
5
4
∫
ℝ
3
ϕ
u
n
u
n
2
d
x
≤
5
4
∫
ℝ
3
ϕ
v
n
v
n
2
d
x
.
Thus, v≠0. By using Fatou’s Lemma and the weakly lower semi-continuous of the norm, we conclude that Pω(v)≤0. Thus, by Lemma 2.2, there exists tv∈(0,1] such that v(x/tv)∈𝒫ω. It follows from (2.3) that
c
ω
≤
I
ω
(
v
(
x
t
v
)
)
=
2
t
v
5
∫
ℝ
3
|
∇
v
|
2
d
x
+
t
v
3
5
∫
ℝ
3
ω
v
2
d
x
+
2
t
v
3
5
∫
ℝ
3
W
(
v
)
d
x
≤
lim inf
n
→
∞
[
2
5
∫
ℝ
3
|
∇
v
n
|
2
d
x
+
1
5
∫
ℝ
3
ω
v
n
2
d
x
+
2
5
∫
ℝ
3
W
(
v
n
)
d
x
]
(2.4)
≤
lim inf
n
→
∞
[
2
5
∫
ℝ
3
|
∇
u
n
|
2
d
x
+
1
5
∫
ℝ
3
ω
u
n
2
d
x
+
2
5
∫
ℝ
3
W
(
u
n
)
d
x
]
=
c
ω
.
Thus, tv=1 and v(x)∈𝒫ω is a minimizer of cω. ∎
Lemma 2.5.
The minimizers of cω are nontrivial ground state solutions to system (1.5).
Proof.
Let u∈𝒫ω be a minimizer of cω, by using the quantitative deformation lemma, we prove Iω′(u)=0 in H-1, where H-1 denotes the dual space of H1(ℝ3).
We argue by contradiction. Suppose that Iω′(u)≠0. Then there exist δ∈(0,1) and α>0 such that ∥Iω′(v)∥H-1>α for each v∈H1(ℝ3) satisfying ∥u-v∥ω≤3δ. Then, by [26, Lemma II.3.2], there exists a pseudo-gradient vector field T in H1(ℝ3) defined on Z, which is a neighborhood of Bδ(u)∩Iωcω, such that for any v∈Z,
∥
T
(
v
)
∥
ω
≤
2
min
{
1
,
∥
I
ω
′
(
u
)
∥
H
-
1
}
,
〈
I
ω
′
(
u
)
,
T
(
v
)
〉
≥
min
{
1
,
∥
I
ω
′
(
u
)
∥
H
-
1
}
∥
I
ω
′
(
u
)
∥
H
-
1
,
where Iωcω:={u∈H1(ℝ3)∣Iω(u)≤cω} and Bδ(u):={v∈H1(ℝ3)∣∥v-u∥ω<δ}.
Choose σ>0 small enough such that ut∈Bδ(u) for all t∈D:=(1-σ,1+σ). Since u∈𝒫ω, it follows from Lemma 2.2 that
(2.5)
c
1
:=
max
∂
D
I
ω
(
u
t
)
<
I
ω
(
u
)
=
c
ω
.
Set ε:=min{(cω-c1)/2,αδ/8}. Let φ be a Lipschitz continuous functional on H1(ℝ3) such that 0≤φ≤1, φ≡1 on Bδ(u)∩Iωcω and φ≡0 on H1(ℝ3)∖Z. Let ξ be a Lipschitz function on ℝ such that 0≤ξ≤1, ξ(l)≡1 if |l-cω|≤ε/2, and ξ≡0 if |l-cω|≥ε. We define
e
(
v
)
:=
{
-
φ
(
v
)
ξ
(
I
ω
(
v
)
)
T
(
v
)
if
v
∈
Z
,
0
if
v
∈
H
1
(
ℝ
3
)
∖
Z
.
Then there exists a global solution η:H1(ℝ3)×[0,+∞)→H1(ℝ3) for the initial value problem
d
d
s
η
(
v
,
s
)
=
e
(
η
(
v
,
s
)
)
,
η
(
v
,
0
)
=
v
.
It is easy to check that
η
(
v
,
1
)
=
v
, if v∉Iω-1([cω-2ε,cω+2ε])∩B2δ(u),
η
(
I
ω
c
ω
+
ε
∩
Z
,
1
)
⊂
I
ω
c
ω
-
ε
,
I
ω
(
η
(
v
,
1
)
)
≤
I
ω
(
v
)
.
Thus, we can conclude that
max
t
∈
D
¯
I
ω
∘
η
(
u
t
,
1
)
<
c
ω
.
We now prove that there exists t0∈D such that η(ut0,1)∈𝒫ω, which contradicts the definition of cω. Define
Φ
0
(
t
)
:=
P
ω
(
u
t
)
,
Φ
1
(
t
)
:=
P
ω
(
η
(
u
t
,
1
)
)
.
It follows from (2.5) that η(ut,1)=ut for t∈∂D. Therefore, by the property of topological degree, we have
deg
(
Φ
1
(
t
)
,
0
,
D
)
=
deg
(
Φ
0
(
t
)
,
0
,
D
)
=
1
,
which implies η(ut0,1)∈𝒫ω for some t0∈D.
Thus, Iω′(u)=0, i.e., u is a nontrivial solution of (1.5). Since Iω(u)=cω, u is a nontrivial ground state solution of (1.5). ∎
Lemma 2.6.
Assume (W4)–(W5) and let u be a nontrivial solution of (1.5). Then u∈L∞(R3)∩C2,γ(R3) for some 0<γ<1 and
lim
|
x
|
→
+
∞
u
(
x
)
=
0
.
Proof.
For T>0, define
u
T
=
{
T
if
u
>
T
,
u
if
u
≤
T
,
-
T
if
u
<
T
.
Multiply (1.5) by |uT|2ru, where r>1 is to be determined later. Since Wu(u)u≥0,
2
r
+
1
(
r
+
1
)
2
∫
|
u
|
≤
T
|
∇
(
|
u
T
|
r
u
)
|
2
d
x
+
∫
|
u
|
>
T
|
∇
(
|
u
T
|
r
u
)
|
2
d
x
≤
1
4
π
∫
ℝ
3
∫
ℝ
3
u
2
(
y
)
|
u
T
(
x
)
|
2
r
u
2
(
x
)
|
x
-
y
|
d
y
d
x
≤
C
(
∫
ℝ
3
|
u
(
y
)
|
12
/
5
d
y
)
5
/
6
(
∫
ℝ
3
|
|
u
T
(
x
)
|
r
u
(
x
)
|
12
/
5
d
y
)
5
/
6
.
Therefore,
∫
ℝ
3
|
∇
(
|
u
T
|
r
u
)
|
2
d
x
≤
C
(
r
+
1
)
(
∫
ℝ
3
|
|
u
T
(
x
)
|
r
u
(
x
)
|
12
/
5
d
y
)
5
/
6
.
By the Sobolev embedding theorem,
(
∫
ℝ
3
|
(
|
u
T
|
r
u
)
|
6
d
x
)
1
/
3
≤
C
1
(
r
+
1
)
(
∫
ℝ
3
|
|
u
T
(
x
)
|
r
u
(
x
)
|
12
/
5
d
y
)
5
/
6
.
Letting T→+∞, we have
(
∫
ℝ
3
|
u
|
6
(
r
+
1
)
d
x
)
1
/
6
(
r
+
1
)
≤
[
C
1
(
r
+
1
)
]
1
/
2
(
r
+
1
)
(
∫
ℝ
3
|
u
|
12
(
r
+
1
)
/
5
d
y
)
5
/
12
(
r
+
1
)
.
Setting rn+1=(5/2)n, we derive
(
∫
ℝ
3
|
u
|
6
⋅
5
n
/
2
n
d
x
)
2
n
/
6
⋅
5
n
≤
exp
(
2
⋅
2
n
5
n
[
ln
C
1
+
n
ln
5
2
]
)
(
∫
ℝ
3
|
u
|
6
⋅
5
n
-
1
/
2
n
-
1
d
x
)
2
n
-
1
/
6
⋅
5
n
-
1
.
Since
∑
n
=
0
∞
2
⋅
2
n
5
n
ln
C
1
+
∑
n
=
0
∞
2
n
⋅
2
n
5
n
ln
5
2
≤
C
2
<
+
∞
,
we deduce
|
u
|
L
∞
(
ℝ
3
)
≤
C
2
|
u
|
L
6
(
ℝ
3
)
.
Since u∈L∞(ℝ3)∩H1(ℝ3), ϕu(x)∈L∞(ℝ3), and it follows that f(x):=-ωu-Wu(u)+ϕuu∈Lq(ℝ3) for all q≥2. Hence, u∈W2,q(ℝ3) by the classical Calderón–Zygmund Lp regularity estimates (see [9, Theorem 9.9]). The Morrey–Sobolev embedding implies u∈𝒞1,α(ℝ3) for some 0<α<1, and by the classical Schauder regularity estimates (see [9, Theorem 4.6]), u∈𝒞2,γ(ℝ3) for some 0<γ<α. Moreover, since f(x)∈Lq(ℝ3) for all q≥2, we can deduce from [9, Theorem 8.17] that
sup
B
1
/
2
(
x
)
|
u
(
x
)
|
≤
C
(
∥
f
∥
L
q
(
B
1
(
x
)
)
+
∥
u
∥
L
2
(
B
1
(
x
)
)
)
.
Combining this with u∈Lq(ℝN) for all q≥2, we can obtain that
lim
|
x
|
→
+
∞
u
(
x
)
=
0
.
Thus, we completed the proof. ∎
Proof of Theorem 1.1.
By Lemma 2.4, there exists a minimizer u of cω. Since Iω and Pω are symmetric functionals, v:=|u| is also a minimizer of cω, and it follows from lemma 2.5 that v is a nonnegative solution of (1.5). Lemma 2.6 implies that v∈L∞(ℝ3)∩𝒞2,γ(ℝ3)∩H1(ℝ3). Then ϕv∈L6(ℝ3)∩𝒞2(ℝ3)∩L∞(ℝ3), therefore
lim
|
x
|
→
∞
ϕ
v
(
x
)
=
0
.
Thus, by the strong maximum principle, we conclude that v(x)>0, i.e., v(x) is a positive ground state solution of system (1.5). ∎
Proof of Theorem 1.2.
Suppose u∈H1(ℝ3) is a nonradial positive ground state solution of system (1.5), and let v be the symmetric-decreasing rearrangement of u. Then, by [18, (iv)–(v) of Chapter 3.3, Theorem 3.7, Theorem 3.9, and Lemma 7.17],
∫
ℝ
3
|
∇
v
|
2
d
x
≤
∫
ℝ
3
|
∇
u
|
2
d
x
,
∫
ℝ
3
W
(
u
)
d
x
=
∫
ℝ
3
W
(
v
)
d
x
,
and
∫
ℝ
3
ϕ
u
u
2
d
x
<
∫
ℝ
3
ϕ
v
v
2
d
x
,
∫
ℝ
3
|
v
n
|
2
d
x
=
∫
ℝ
3
|
u
n
|
2
d
x
.
The last strict inequality and Lemma 2.3 imply that there exists a tv∈(0,1) such that v(x/tv)∈𝒫ω. Then, similarly to (2), we get a contradiction.
Finally, it follows from Lemma 2.6 and the comparison principle (see [9, Theorem 3.3]) that there exists R>0 such that
|
u
(
x
)
|
≤
C
e
-
δ
|
x
|
for
|
x
|
>
R
,
for some positive constants C, δ. ∎
3 The Nonautonomous Case
In this section, we study the nonautonomous Choquard system (1.6).
First, we give some notation. Define
H
:=
{
u
∈
H
1
(
ℝ
3
)
|
∫
ℝ
3
V
(
x
)
u
2
𝑑
x
<
+
∞
}
with the norm
∥
u
∥
2
=
∫
ℝ
3
|
∇
u
|
2
+
V
(
x
)
u
2
d
x
.
The energy functional corresponding to (1.6) is defined by
I
V
(
u
)
=
1
2
∫
ℝ
3
|
∇
u
|
2
+
V
(
x
)
u
2
d
x
+
∫
ℝ
3
W
(
u
)
d
x
-
1
4
∫
ℝ
3
ϕ
u
u
2
d
x
,
u
∈
H
.
Obviously, I∈𝒞1(H,ℝ) and nontrivial critical points are nontrivial solutions of system (1.6). By a similar argument as that in Lemma 2.6, we can deduce that each weak solution to system (1.6) is a classical solution.
Since there is a competitive interplay of the nonlinearities ∫ℝ3W(u)𝑑x and ∫ℝ3ϕuu2𝑑x, it is difficult to get the boundedness of any (PS) sequence even if the (PS) sequence has been obtained. To overcome this difficulty, we need the following result due to Jeanjean [12].
Proposition 3.1 (See [12, Theorem 1.1]).
Let (X,∥⋅∥) be a Banach space and let T⊂R+ be an interval. Consider a family of C1 functionals on X of the form
Φ
λ
(
u
)
=
A
(
u
)
-
λ
B
(
u
)
for all
λ
∈
T
,
with B(u)≥0, and either A(u)→+∞ or B(u)→+∞ as ∥u∥→+∞. Assume that there are two points v1, v2∈X such that
c
λ
:=
inf
γ
∈
Γ
max
t
∈
[
0
,
1
]
Φ
λ
(
γ
(
t
)
)
>
max
{
Φ
λ
(
v
1
)
,
Φ
λ
(
v
2
)
}
for all
λ
∈
T
,
where
Γ
=
{
γ
∈
C
(
[
0
,
1
]
,
X
)
∣
γ
(
0
)
=
v
1
,
γ
(
1
)
=
v
2
}
.
Then, for almost every λ∈T, there is a bounded (PS)cλ sequence in X.
Assume that (V1)–(V3) hold. We will apply Proposition 3.1 to prove Theorem 1.3.
Set T=[δ,1], where δ∈(0,1) is a positive constant. For λ∈[δ,1], we consider a family of functionals on H defined by
I
V
,
λ
(
u
)
=
1
2
∫
ℝ
3
|
∇
u
|
2
+
V
(
x
)
u
2
d
x
+
∫
ℝ
3
W
(
u
)
d
x
-
λ
4
∫
ℝ
3
ϕ
u
u
2
d
x
.
Then IV,λ(u)=A(u)-λB(u), where
A
(
u
)
=
1
2
∫
ℝ
3
|
∇
u
|
2
+
V
(
x
)
u
2
d
x
+
∫
ℝ
3
W
(
u
)
d
x
→
+
∞
as
∥
u
∥
→
+
∞
,
and
B
(
u
)
=
1
4
∫
ℝ
3
ϕ
u
u
2
𝑑
x
≥
0
.
Lemma 3.2.
Assume that (V1)–(V3) hold. Then
there exists
v
∈
H
∖
{
0
}
such that
I
V
,
λ
(
v
)
≤
0
for all
λ
∈
[
δ
,
1
]
,
we have
c
λ
:=
inf
γ
∈
Γ
max
t
∈
[
0
,
1
]
I
V
,
λ
(
γ
(
t
)
)
>
max
{
I
V
,
λ
(
0
)
,
I
V
,
λ
(
v
)
}
for all
λ
∈
[
δ
,
1
]
,
where
Γ
=
{
γ
∈
C
(
[
0
,
1
]
,
H
)
∣
γ
(
0
)
=
0
,
γ
(
1
)
=
v
}
.
Proof.
(i) For fixed u∈H1(ℝ3)∖{0} and any λ∈[δ,1], we have that
I
V
,
λ
(
u
)
≤
I
δ
∞
(
u
)
:=
1
2
∫
ℝ
3
|
∇
u
|
2
+
V
∞
u
2
d
x
+
∫
ℝ
3
W
(
u
)
d
x
-
δ
4
∫
ℝ
3
ϕ
u
u
2
d
x
.
Obviously, Iδ∞(ut)→-∞ as t→+∞. Hence, setting v=ut for t large, we have IV,λ(v)≤Iδ∞(v)<0.
(ii) Since
I
V
,
λ
(
u
)
≥
1
2
∫
ℝ
3
|
∇
u
|
2
+
V
(
x
)
u
2
d
x
-
C
∥
u
∥
4
,
0 is the strict local minimum point of IV,λ, hence cλ>0. ∎
It follows from Lemma 3.2 and the definition of IV,λ(u) that IV,λ(u) satisfies the assumptions of Proposition 3.1 with X=H and Φλ=IV,λ. So, for a.e. λ∈[δ,1], there exists a bounded sequence {un}⊂H (for simplicity, we denote {un} instead of {un(λ)}) such that
I
V
,
λ
(
u
n
)
→
c
λ
,
I
V
,
λ
′
(
u
n
)
→
0
in
H
*
as
n
→
∞
,
where H* denotes the dual space of H.
By Theorem 1.1, we know that for each λ∈[δ,1], the associated limit problem
{
-
Δ
u
+
V
∞
u
+
W
u
(
u
)
=
λ
ϕ
(
x
)
u
,
x
∈
ℝ
3
,
-
Δ
ϕ
=
u
2
,
x
∈
ℝ
3
,
has a positive ground state solution in H1(ℝ3), i.e., for any λ∈[δ,1],
c
λ
∞
:=
inf
u
∈
𝒫
λ
∞
I
λ
∞
(
u
)
is achieved by some uλ∞∈𝒫λ∞:={u∈H1(ℝ3)∖{0}∣Pλ∞(u)=0}, where
P
λ
∞
(
u
)
:=
1
2
∫
ℝ
3
|
∇
u
|
2
d
x
+
3
2
∫
ℝ
3
V
∞
u
2
d
x
+
3
∫
ℝ
3
W
(
u
)
d
x
-
5
λ
4
∫
ℝ
3
ϕ
u
u
2
d
x
.
Lemma 3.3.
Assume that (V1)–(V3) hold. Then cλ<cλ∞ for any λ∈[δ,1].
Proof.
Let uλ∞ be the minimizer of cλ∞. By Lemma 2.2, we have that Iλ∞(uλ∞)=maxt>0Iλ∞(uλ∞(x/t)). Then we choose v=uλ∞(x/t¯) for t¯ large as in Lemma 3.2 (i). By (V2), for any λ∈[δ,1], we can deduce that
c
λ
≤
max
t
>
0
I
V
,
λ
(
u
λ
∞
(
x
t
)
)
<
max
t
>
0
I
λ
∞
(
u
λ
∞
(
x
t
)
)
=
I
λ
∞
(
u
λ
∞
)
=
c
λ
∞
.
∎
In order to prove that the functional IV,λ satisfies the (PS)cλ condition for a.e. λ∈[δ,1], we need the following global compactness lemma.
Lemma 3.4.
Assume that (V1)–(V3) and (W4)–(W5) hold. For c>0 and for any λ∈[δ,1], let {un}⊂H be a bounded (PS)c sequence for IV,λ Then there exists a u∈H1(R3) such that IV,λ′(u)=0, and either
there exist
l
∈
ℕ
, {ynk}⊂ℝ3, with |ynk|→∞ as n→∞ for each 1≤k≤l, and nontrivial solutions w1,w2,…,wl of the following problem:
(3.1)
{
-
Δ
u
+
V
∞
u
+
W
u
(
u
)
=
λ
ϕ
(
x
)
u
,
x
∈
ℝ
3
,
-
Δ
ϕ
=
u
2
,
x
∈
ℝ
3
,
such that
c
=
I
V
,
λ
(
u
)
+
∑
k
=
1
l
I
λ
∞
(
w
k
)
and
∥
u
n
-
u
-
∑
k
=
1
l
w
k
(
⋅
+
y
n
k
)
∥
→
0
.
Proof.
Since {un}⊂H is a bounded (PS)c sequence for IV,λ, i.e.,
I
V
,
λ
(
u
n
)
→
c
,
I
V
,
λ
′
(
u
n
)
→
0
in
H
*
as
n
→
∞
,
there exists u∈H such that
u
n
⇀
u
weakly in
H
.
Furthermore, IV,λ′(un)→0 implies that
∫
ℝ
3
(
∇
u
∇
φ
+
V
(
x
)
u
φ
)
𝑑
x
+
∫
ℝ
3
W
u
(
u
)
φ
𝑑
x
-
∫
ℝ
3
ϕ
u
u
φ
𝑑
x
=
0
for
φ
∈
H
.
That is IV,λ′(u)=0.
We next show that either (i) or (ii) holds. The argument is standard, and for the reader’s convenience, we give a detailed proof. Step 1. Setting un1:=un-u, by [22, Lemma 2.4], we have that
∫
ℝ
3
ϕ
u
n
u
n
2
d
x
=
∫
ℝ
3
ϕ
u
n
1
|
u
n
1
|
2
d
x
+
∫
ℝ
3
ϕ
u
u
2
d
x
+
o
n
(
1
)
.
Thus, one can easily check that
∥
u
n
1
∥
2
=
∥
u
n
∥
2
-
∥
u
∥
2
+
o
n
(
1
)
,
∫
ℝ
3
W
(
u
n
1
)
=
∫
ℝ
3
W
(
u
n
)
-
∫
ℝ
3
W
(
u
)
+
o
n
(
1
)
,
I
λ
∞
(
u
n
1
)
=
c
-
I
V
,
λ
(
u
)
+
o
n
(
1
)
,
(
I
λ
∞
)
′
(
u
n
1
)
→
0
in H*.
Define
δ
1
=
lim inf
n
→
∞
sup
x
∈
ℝ
3
∫
B
1
(
y
)
|
u
n
1
|
2
d
x
.
Vanishing: If δ1=0, then, un1→0 strongly in Ls(ℝ3) for 2<s<6. Since (Iλ∞)′(un1)→0 in H*, we can deduce that un1→0 strongly in H and the proof is completed.
Non-vanishing: If δ1>0, then there exists a sequence {yn1}⊂ℝ3 such that
∫
B
1
(
y
n
1
)
|
u
n
1
|
2
d
x
>
1
2
δ
1
.
Set wn1:=un1(⋅+yn1). Obviously, {wn1} is bounded in H and we may assume that wn1⇀w1 in H. Hence, (Iλ∞)′(w1)=0. Since
∫
B
1
(
0
)
|
w
n
1
|
2
d
x
=
∫
B
1
(
y
n
1
)
|
u
n
1
|
2
d
x
>
1
2
δ
1
,
we deduce that w1≠0. Moreover, un1⇀0 in H implies that {yn1} is unbounded. Hence, we may assume that |yn1|→∞.
Step 2. Set un2:=un-u-w1(⋅-yn1). We can similarly check that
∫
ℝ
3
ϕ
u
n
u
n
2
d
x
=
∫
ℝ
3
ϕ
u
n
2
|
u
n
2
|
2
d
x
+
∫
ℝ
3
ϕ
u
u
2
d
x
+
∫
ℝ
3
ϕ
w
1
|
w
1
|
2
d
x
+
o
n
(
1
)
and
∥
u
n
2
∥
2
=
∥
u
n
∥
2
-
∥
u
∥
2
-
∥
w
n
1
∥
2
+
o
n
(
1
)
,
∫
ℝ
3
W
(
u
n
2
)
=
∫
ℝ
3
W
(
u
n
)
-
∫
ℝ
3
W
(
u
)
-
∫
ℝ
3
W
(
w
n
1
)
+
o
n
(
1
)
,
I
λ
∞
(
u
n
2
)
=
c
-
I
V
,
λ
(
u
)
-
I
λ
∞
(
w
n
1
)
+
o
n
(
1
)
,
(
I
λ
∞
)
′
(
w
n
2
)
→
0
in H*.
Similar to the arguments in Step 1, set
δ
2
=
lim inf
n
→
∞
sup
x
∈
ℝ
3
∫
B
1
(
y
)
|
u
n
2
|
2
d
x
.
If vanishing occurs, then ∥un2∥→0, i.e., ∥un-u-w1(⋅+yn1)∥→0. Moreover, by (2a) and (2c),
c
=
I
V
,
λ
(
u
)
+
I
λ
∞
(
w
1
)
.
If non-vanishing occurs, then there exists a sequence {yn2}⊂ℝ3 and a nontrivial w2∈H such that wn2:=un2(⋅+yn2)⇀w2 weakly in H. Then, by (2d), we have that (Iλ∞)′(w2)=0. Furthermore, un2⇀0 weakly in H implies that |yn2|→∞ and |yn2-yn1|→∞.
We next proceed by iteration. Recall that if wk is a nontrivial critical point of Iλ∞, then Iλ∞(wk)>0. So there exists some finite number l∈ℕ such that only the vanishing case occurs after Step 1. Then the lemma is proved. ∎
Lemma 3.5.
Assume that (V1)–(V3) and (W4)–(W5) hold. For λ∈[δ,1], let {un}⊂H be a bounded (PS)cλ sequence for IV,λ. Then there exists a uλ∈H such that
u
n
→
u
λ
strongly in
H
.
Proof.
By Lemma 3.4, for λ∈[δ,1], there exists uλ∈H such that
u
n
⇀
u
λ
weakly in
H
,
I
V
,
λ
′
(
u
λ
)
=
0
and either (i) or (ii) occurs.
If (ii) occurs, i.e., there exist l∈ℕ, {ynk}⊂ℝ3, with |ynk|→∞ as n→∞ for each 1≤k≤l, and nontrivial solutions w1,…,wk of problem (3.1) such that
(3.2)
c
λ
=
I
V
,
λ
(
u
λ
)
+
∑
k
=
1
l
I
λ
∞
(
w
k
)
and
(3.3)
∥
u
n
-
u
λ
-
∑
k
=
1
l
(
w
k
(
⋅
+
y
n
k
)
)
∥
→
0
strongly in
H
.
Since uλ is a solution of system (1.6), uλ satisfies the following Pohozaev identity:
∫
ℝ
3
|
∇
u
λ
|
2
d
x
+
3
∫
ℝ
3
V
(
x
)
u
λ
2
d
x
+
∫
ℝ
3
〈
∇
V
(
x
)
,
x
〉
u
λ
2
d
x
+
6
∫
ℝ
3
W
(
u
λ
)
d
x
=
5
λ
2
∫
ℝ
3
ϕ
u
λ
u
λ
2
d
x
.
Thus,
I
V
,
λ
(
u
λ
)
=
2
5
∫
ℝ
3
|
∇
u
λ
|
2
d
x
+
1
5
∫
ℝ
3
V
(
x
)
u
λ
2
d
x
-
1
10
∫
ℝ
3
〈
∇
V
(
x
)
,
x
〉
u
λ
2
d
x
+
2
5
∫
ℝ
3
W
(
u
λ
)
d
x
.
Since cλ∞ is the ground state energy of equation (3.1), it follows form (3.2)–(3.3) that
c
λ
=
I
V
,
λ
(
u
λ
)
+
∑
k
=
1
l
I
λ
∞
(
w
k
)
≥
I
V
,
λ
(
u
λ
)
+
l
c
λ
∞
≥
c
λ
∞
.
That is cλ≥cλ∞, which contradicts Lemma 3.3. So (i) holds, i.e., un→uλ strongly in H. Thus, uλ is a nontrivial critical point of IV,λ and IV,λ(uλ)=cλ. ∎
Proof of Theorem 1.3.
We complete the proof in two steps.
Step 1. By Proposition 3.1 and Lemma 3.5, for a.e. λ∈[δ,1], there exists a nontrivial critical point uλ∈H of IV,λ with IV,λ(uλ)=cλ.
Choosing a sequence {λn}⊂[δ,1] satisfying λn→1, we have a sequence of nontrivial critical points {uλn} of IV,λn and IV,λn(uλn)=cλn. Therefore,
1
2
∫
ℝ
3
|
∇
u
λ
n
|
2
d
x
+
3
2
∫
ℝ
3
V
(
x
)
u
λ
n
2
d
x
+
1
2
∫
ℝ
3
〈
∇
V
(
x
)
,
x
〉
u
λ
n
2
d
x
+
3
∫
ℝ
3
W
(
u
λ
n
)
d
x
=
5
λ
n
4
∫
ℝ
3
ϕ
λ
n
u
λ
n
2
d
x
and
1
2
∫
ℝ
3
|
∇
u
λ
n
|
2
d
x
+
1
2
∫
ℝ
3
V
(
x
)
u
λ
n
2
d
x
+
∫
ℝ
3
W
(
u
λ
n
)
d
x
-
λ
n
4
∫
ℝ
3
ϕ
u
λ
n
u
λ
n
2
d
x
=
c
λ
n
.
By (V3), we then conclude that {uλn} is bounded in H. Moreover, by [12, Lemma 2.3],
lim
n
→
∞
I
V
,
1
(
u
λ
n
)
=
lim
n
→
∞
(
I
V
,
λ
n
(
u
λ
n
)
+
λ
n
-
1
4
∫
ℝ
3
ϕ
u
λ
n
u
λ
n
2
d
x
)
=
lim
n
→
∞
c
λ
n
=
:
c
1
and
lim
n
→
∞
〈
I
V
,
1
′
(
u
λ
n
)
,
φ
〉
=
lim
n
→
∞
(
〈
I
V
,
λ
n
′
(
u
λ
n
)
,
φ
〉
+
(
λ
n
-
1
)
∫
ℝ
3
ϕ
u
λ
n
u
λ
n
φ
𝑑
x
)
=
0
.
Obviously, {uλn} is a bounded (PS)c1 sequence for IV=IV,1. Then, by Lemma 3.5, there exists a nontrivial critical point u1∈H for IV with IV(u1)=c1. Step 2. Next, we prove the existence of a ground state solution for problem (1.6). Set
m
:=
inf
{
I
V
(
u
)
∣
u
∈
H
∖
{
0
}
,
I
V
′
(
u
)
=
0
}
.
Then, by (V3), we can deduce that 0<m≤IV(u1)=c1. Let {un} be a sequence of nontrivial critical points of IV satisfying IV(un)→m. Using the same arguments as in Step 1, we can deduce that {un} is bounded in H1(ℝ3), i.e., {un} is a bounded (PS)m sequence of IV. Similar to the arguments in Lemma 3.5, there exists a nontrivial u∈H such that IV(u)=m and IV′(u)=0. By similar arguments as in the proof of Lemma 2.6 and Theorem 1.1, we conclude that u is a positive ground state solution for problem (1.6). Thus, the proof is completed. ∎
