# Existence of Multiple Positive Solutions for Choquard Equation with Perturbation

Authors: Tao Xie; Lu Xiao; Jun Wang
Journal: Advances in Mathematical Physics
 (2015)

---

## Abstract

This paper is concerned with the following Choquard equation with perturbation:-
Δ
u
+
V
(
x
)
u
=
(
1
/
|
x
|
α
∗
|
u
|
p
)
|
u
|
p
-
2
u
+
g
(
x
), u
∈
H
1
(
R
N
), where N
≥
3, α
∈
(
0
,
N
), and 2
-
(
α
/
N
)
<
p
<
(
2
N
-
α
)
/
(
N
-
2
). This kind of equations is well known as the Choquard or nonlinear Schrödinger-Newton equation. Under some assumptions for the functions V
(
x
), we prove the existence of multiple positive solutions of the equation. Moreover, we also show that these results still hold for more generalized Choquard equation with perturbation.


---

## Body

## 1. Introduction and Main Results

In present paper, we are concerned with the existence of multiple solutions to the following Choquard equation with perturbation:(1)
-
Δ
u
+
V
x
u
=
1
x
α
∗
u
p
u
p
-
2
u
+
g
x
,
x
∈
R
N
,where N
≥
3, α
∈
(
0
,
N
), 2
-
(
α
/
N
)
<
p
<
(
2
N
-
α
)
/
(
N
-
2
), and V is nonnegative function.This kind of (1) arises in various physical contexts, especially in the case where N
=
3, α
=
2, p
=
2, and g
=
0. Then (1) becomes(2)
-
Δ
u
+
V
x
u
=
1
x
∗
u
2
u
,
x
∈
R
3
.It is called the stationary nonlinear Choquard equation or the nonlinear Schrödinger-Newton equation. In general, many mathematicians are concerned with the positive solitary solutions of the following nonlinear generalized Choquard equation:(3)
i
φ
t
-
Δ
φ
+
K
x
φ
-
1
x
α
∗
φ
p
φ
p
-
2
φ
=
0
,
x
∈
R
N
,
t
>
0
,where the powers p
≥
2 and α
∈
(
0
,
N
). In order to obtain the solitary solutions of (3), we set φ
(
x
,
t
)
=
e
i
ω
t
u
(
x
) (ω
>
0 is a constant) in (3) and get the stationary equation of (1) without perturbation, where V
(
x
)
=
K
(
x
)
-
ω.In 1954, paper [1] proposed model (2) to the description of the quantum theory of a polaron. Later, (2) was proposed by Choquard in 1976 as an approximation to Hartree-Fock theory for one component plasma [2]. In the 1990s the same equation reemerged as a model of self-gravitating matter [3, 4] and is known in that context as the Schrödinger-Newton equation. In recent years, many papers are concerned with the existence of solutions of (3). Lieb [2] proved the existence and uniqueness of the ground state to (2). Lions [5] obtained the existence of a sequence of radially symmetric solutions for (2) by using variational methods. Papers [6, 7] proved the existence of multibump solutions of (2). Paper [8] proved that every positive solution of (2) is radially symmetric and monotone decreasing about some point by using moving plane methods. Furthermore, the authors obtained the uniqueness of positive solutions for (2). Clapp and Salazar [9] proved the existence of positive and sign changing solutions of (2) when R
3 and u are replaced by an exterior bounded domain Ω and W
(
x
)
u, respectively. Moroz and Van Schaftingen [10] showed the regularity, positivity, and radial symmetry of the ground states for the optimal range of parameters, and they also obtained decay asymptotic at infinity for these ground states. The more general system (3) was considered in [11]. Moroz and Van Schaftingen [12] obtained the nonexistence and optimal decay of supersolutions of (3). Cingolani and Secchi [13] considered the existences of ground states for the pseudorelativistic Hartree equation. For semiclassical cases, the existence of multiple semiclassical solutions was considered in [14]. Paper [15] considered the existence of semiclassical regime of standing wave solutions of a Schrödinger equation in presence of nonconstant electric and magnetic potentials. Cingolani and Secchi [16] studied the semiclassical limit for the pseudorelativistic Hartree equation. Under the assumptions on the decay of V, paper [17] proved the existence of positive solutions by using variational methods and nonlocal penalization technique.Motivated by the works we mentioned above, in this paper we study the existence of multiple solutions to the nonlinear Choquard equation with perturbation. This kind of problems is often referred to as being nonlocal because of the appearance of the term∫
R
N
‍
∫
R
N
‍
u
x
p
u
y
p
/
x
-
y
α
d
x
d
y in the energy functional. This leads to the fact that (1) is no longer a pointwise identity. The main difficulties when dealing with this problem lie in the presence of the nonlocal term and the lack of compactness due to the unboundedness of the domain R
N. Under some conditions on V, in the present paper we recover the compactness and find two nontrivial solutions of (1) by using variational methods.In what follows, we assume thatV
∈
C
(
R
N
,
R
+
) and satisfies the following condition:(
V
0)
V is coercive; that is, l
i
m
|
x
|
→
∞
V
(
x
)
=
+
∞.A solution is called a ground state solution (or positive ground state solution) if its energy is minimal among all the nontrivial solutions (or all the nontrivial positive solutions) of (1). A bound state solution refers to limited-energy solution.Then, we have the following main results.Theorem 1.
Assume that(
V
0
) holds. Then there exists ɛ
∗
>
0 such that, for every g
∈
L
p
/
p
-
1
(
R
N
) with |
g
|
p
/
p
-
1
≤
ɛ
∗, problem (1) admits two nontrivial solutions. Furthermore, if g is positive, problem (1) has one positive ground state solution and one positive bound state solution.Remark 2.
As we will see later, condition(
V
0
) guarantees that the embedding E
↪
L
p
(
R
N
) is compact (see [18]), where E
=
{
u
∈
D
1,2
(
R
N
)
:
∫
R
N
‍
V
(
x
)
u
2
<
∞
}. So, similar to [19], condition (
V
0
) can be replaced by the following general condition:(V
0
′)
There existsr
>
0 such that, for any k
>
0,(4)
lim
y
→
∞
⁡
m
x
∈
R
N
:
V
x
≤
k
∩
B
r
y
=
0
,
whereB
r
(
y
)
=
{
x
∈
R
N
:
|
x
-
y
|
<
r
} and m is the Lebesgue measure in R
N.


## 2. Variational Setting

Throughout the paper, we use the following notations:(i)
· is the norm in H
1
(
R
N
) defined by u
2
=
∫
R
N
‍
(
∇
u
2
+
u
2
).
(ii)
·
p is the norm in L
p
(
R
N
) defined by u
p
=
∫
R
N
‍
u
p
1
/
p for 0
<
p
<
∞.
(iii)
Letc and C
i be some positive numbers.The main purpose of this section is to establish the variational setting for problem (1). We first recall the following classical Hardy-Littlewood-Sobolev inequality (see [20, Theorem 4.3]).Lemma 3.
Assume thatf
∈
L
p
(
R
N
) and h
∈
L
q
(
R
N
). Then one has (5)
∫
R
N
∫
R
N
f
x
h
y
x
-
y
α
d
x
d
y
≤
c
p
,
q
,
α
f
p
h
q
,where 1
<
p, q
<
∞, 0
<
α
<
N, and 1
/
p
+
1
/
q
+
α
/
N
=
2.Define the Sobolev space(6)
E
=
u
∈
D
1,2
R
N
:
∫
R
N
V
x
u
2
<
∞with the norm(7)
u
E
2
=
∫
R
N
∇
u
2
+
V
x
u
2
.This is a Hilbert space and its inner product is denoted by (
·
,
·
)
E. It is easy to check that the embedding E
↪
H
1
(
R
N
) is continuous. Moreover, under condition (
V
0
), we infer that the embedding E
↪
L
t
(
R
N
)
(
∀
t
∈
(
2
,
2
N
/
N
-
2
)
) is compact (see, e.g., [18]).By Lemma3, the functional(8)
B
u
=
∫
R
N
∫
R
N
u
x
p
u
y
p
x
-
y
α
d
x
d
yis well defined if |
u
|
p
∈
L
s
(
R
N
) for s
>
1 defined by(9)
2
s
+
α
N
=
2
.Therefore, since u
∈
E, it follows that s
p
∈
[
2
,
2
N
/
N
-
2
]. Here we only consider the subcritical case. So we have(10)
2
-
α
N
<
p
<
2
N
-
α
N
-
2
.The energy functional corresponding to system (1) is defined by(11)
J
u
=
1
2
∫
R
N
∇
u
2
+
V
x
u
2
-
1
2
p
∫
R
N
∫
R
N
u
x
p
u
y
p
x
-
y
α
d
x
d
y
-
∫
R
N
g
u
,for u
∈
E. From (8)–(10) and g
∈
L
p
/
p
-
1
(
R
N
), we infer that J is well defined in E. Furthermore, J
∈
C
2
(
E
,
R
), and a critical point of J is a solution of (1).In order to prove Theorem1 we will constrain the functional J on the set(12)
N
=
u
∈
E
:
J
′
u
,
u
=
0
.Usually, this set is called Nehari manifold. It is well-known that critical points of J lie in the Nehari manifold. Denote Φ
u
=
J
′
u
,
u. Thus, we know that(13)
J
′
u
,
u
=
u
E
2
-
∫
R
N
∫
R
N
u
x
p
u
y
p
x
-
y
α
d
x
d
y
-
∫
R
N
g
u
,
Φ
′
u
,
u
=
2
u
E
2
-
2
p
∫
R
N
∫
R
N
u
x
p
u
y
p
x
-
y
α
d
x
d
y
-
∫
R
N
g
u
.In order to prove the existence of multiple nontrivial solutions for (1), we will divide the Nehari manifold N into the following three parts:(14)
N
+
=
u
∈
N
:
Φ
′
u
,
u
>
0
,
N
0
=
u
∈
N
:
Φ
′
u
,
u
=
0
,
N
-
=
u
∈
N
:
Φ
′
u
,
u
<
0
.Obviously, only N
0 contains the element 0. Furthermore, it is easy to see that N
+
∪
N
0 and N
-
∪
N
0 are both closed subsets of E.Next we will give some explanation for the partition of Nehari manifoldsN. Set(15)
A
=
A
u
=
u
E
2
,
B
=
B
u
=
∫
R
N
∫
R
N
u
x
p
u
y
p
x
-
y
α
d
x
d
y
,
C
=
C
u
=
∫
R
N
g
u
.We define the fibering map(16)
ϕ
t
=
ϕ
u
t
=
J
t
u
=
A
2
t
2
-
B
2
p
t
2
p
-
C
t
,
∀
t
>
0
.Thus,(17)
ϕ
′
t
=
A
t
-
B
t
2
p
-
1
-
C
,
ϕ
′
′
t
=
A
-
2
p
-
1
B
t
2
p
-
2
.Obviously, t
u
∈
N with t
>
0 if and only if ϕ
′
t
=
0. It is well-known that if the function ϕ
(
t
) has unique global maximum point, then the set N is homotopic to unit ball of E. Moreover, the set N is a natural constraint for the functional J. This means that if the infimum is attained by u
∈
N, then u is a solution of (1). However, in our situation, the global maximum point of ϕ is not unique. This leads us to partition the set N according to the critical points of ϕ. This kind of idea was first introduced by Tarantello in [21]. Later, many mathematicians apply this idea to study other problems; for instance, see [22–24] and the references therein.Ifϕ
′
′
t
0
=
0 we know that(18)
t
0
=
A
2
p
-
1
B
1
/
2
p
-
2
>
0
,
for
p
>
2
-
α
N
>
1
.So if C
>
0 small, we infer that ϕ
′
t
=
0 has two positive solutions t
1
<
t
0
<
t
2. Moreover, according to the sign of second derivative of the fibering map at these points, one knows that t
1, t
2, and t
0 are local minimum, local maximum, and turning point, respectively. Hence, the sets N
± and N
0 contain the local minimum, local maximum, and turning point, respectively. Furthermore, one can check that(19)
t
u
∈
N
+
and
t
>
0
⟺
ϕ
′
t
=
0
and
ϕ
′
′
t
>
0
,
t
u
∈
N
0
and
t
>
0
⟺
ϕ
′
t
=
0
and
ϕ
′
′
t
=
0
,
t
u
∈
N
-
and
t
>
0
⟺
ϕ
′
t
=
0
and
ϕ
′
′
t
=
0
.Now we are ready to study the properties of sets N
± and N
0.Lemma 4.
Assume that the nonzero functiong
∈
L
p
/
p
-
1
(
R
N
). Then the following results hold.(i)
There exists positive constantɛ
1 such that N
0
=
{
0
} when |
g
|
p
/
p
-
1
<
ɛ
1. Furthermore, N
±
≠
∅.
(ii)
N
- is closed.Proof.
(i) In order to prove thatN
0
=
{
0
}, we only need to show that, for u
∈
E
∖
{
0
}, ϕ
(
t
) has no critical point that is a turning point. Without loss of generality we can assume that u
E
=
1. Define(20)
k
t
=
A
t
-
B
t
2
p
-
1
.Then ϕ
(
t
)
=
k
(
t
)
-
C. Let us consider the graph of k
(
t
). Since k
′
′
t
=
-
B
(
2
p
-
1
)
(
2
p
-
2
)
t
2
p
-
3
<
0 for t
>
0, k
(
t
) is strictly concave. Moreover, l
i
m
t
→
0
+
k
(
t
)
=
0, l
i
m
t
→
-
∞
k
(
t
)
=
-
∞, and k
(
t
)
>
0 for t
>
0 small. Thus, we know that k
(
t
) has unique global maximum point t
0 and k
(
t
0
)
=
2
A
(
p
-
1
)
/
2
p
-
1
A
/
(
2
p
-
1
)
B
1
/
2
p
-
1
:
=
k
0. So we infer from (16) and (17) that if 0
<
C
<
k
0, the equation ϕ
′
t
=
0 has exactly two roots t
1
,
t
2 satisfying t
1
<
t
0
<
t
2. If C
≤
0, the equation ϕ
′
t
=
0 has exactly one root t
~
1
>
t
0. Since ϕ
′
′
(
t
)
=
A
-
(
2
p
-
1
)
B
t
2
p
-
2, it follows that ϕ
′
′
t
1
>
0, ϕ
′
′
t
2
<
0, and ϕ
′
′
(
t
~
1
)
<
0. Hence, if 0
<
C
<
k
0, we know that t
1
u
∈
N
+, t
2
u
∈
N
-. If C
≤
0, we know that t
~
1
u
∈
N
-. Also, since the sets {
u
∈
E
:
u
E
=
1
,
0
<
C
<
k
0
} and {
u
∈
E
:
u
E
=
1
,
C
≤
0
} are nonempty, we infer that N
±
≠
∅. This also implies that N
0
=
{
0
}.
Next we still need to findɛ
1
>
0 such that for |
g
|
p
/
p
-
1
<
ɛ
1
(21)
C
<
k
0
=
2
A
p
-
1
2
p
-
1
A
2
p
-
1
B
1
/
2
p
-
1
.Since A
=
1 and u lies on the unit sphere of E, we infer from Lemma 3 that B has upper bound. So there exists ɛ
2
>
0 such that(22)
0
<
ɛ
2
≤
2
p
-
1
2
p
-
1
1
2
p
-
1
sup
u
E
=
1
⁡
B
u
1
/
2
p
-
1
.Furthermore, one deduces from Sobolev inequality and Hölder inequality that(23)
C
=
C
u
=
∫
R
N
g
u
≤
g
p
/
p
-
1
u
p
≤
S
p
g
p
/
p
-
1
u
E
=
S
p
g
p
/
p
-
1
,where S
p denotes the best constant satisfying S
p
u
E
≥
|
u
|
p. Combining (22) and (23) we know that if ɛ
1
=
ɛ
2
/
S
p, then (21) holds.
(ii) Sincec
l
(
N
-
)
⊂
N
-
∪
{
0
}, it suffices to show that 0
∉
c
l
(
N
-
), where c
l
(
N
-
) denotes the closure of N
-. Equivalently, we only need to prove that d
i
s
t
(
0
,
N
-
)
>
0. For any u
∈
N
-, we denote u
0
=
u
/
u
E. Then u
0
E
=
1. From the proof of part (i) we know that if C
≤
0, the equation ϕ
′
(
t
)
=
0 has exactly one root t
~
1
>
t
0 such that t
~
1
u
0
∈
N
-. Thus, t
~
1
u
0
=
u
∈
N
-, and so t
~
1
=
u
E. If 0
<
C
<
k
0, the equation ϕ
′
t
=
0 has exactly two roots t
1
,
t
2 satisfying t
1
<
t
0
<
t
2, which satisfy t
1
u
0
∈
N
+ and t
2
u
0
∈
N
-. Hence, t
2
u
0
=
u
∈
N
- and so t
2
=
u
E. Anyway, we infer from t
2
,
t
~
1
>
t
0 that u
E
>
t
0. Since A
=
1 and B is bounded from above, it follows from (22) that there exists σ
0
>
0 such that u
E
>
σ
0. That is, (24)
dist
⁡
0
,
N
-
=
inf
u
∈
N
-
⁡
u
E
≥
σ
0
>
0
.This ends the proof of the lemma.Now we define the minimization problems:(25)
c
+
=
inf
u
∈
N
+
⁡
J
u
,
c
-
=
inf
u
∈
N
-
⁡
J
u
.We will prove that if the infimum in (25) is attained by u, then u is a solution of (1). That is, the following results hold.Lemma 5.
Ifc
+ or c
- is attained by u
0
∈
N
+ or u
0
∈
N
-, then u
0 is a nontrivial solution of (1). Moreover, c
+
<
0
<
c
- for |
g
|
p
/
p
-
1
<
ɛ
2, where ɛ
2
>
0 is a small positive number.Proof.
Assume thatu
0
∈
N
+ such that J
(
u
0
)
=
c
+. We define U
=
{
u
∈
E
:
〈
Φ
′
u
,
u
〉
>
0
}. Thus,(26)
N
+
=
u
∈
U
:
Φ
u
=
0
.Hence, by using [25, Theorem 4.1.1] we infer that there exists Lagrangian multiplier l
∈
R such that(27)
J
′
u
0
=
l
Φ
′
u
0
.Moreover, it follows from J
′
u
0
,
u
0
=
0 and Φ
′
u
0
,
u
0
>
0 that l
=
0. Thus, u
0 is a nontrivial solution of (1). Similarly, if c
- is attained by u
0
∈
N
- we can also prove that u
0 is a nontrivial solution of (1).
Next we prove the latter part of the lemma. As in the proof of Lemma4, we can take u
∈
E such that u
E
=
1 and(28)
0
<
C
<
t
0
=
A
2
p
-
1
B
1
/
2
p
-
2
.The equation ϕ
′
t
=
0 has exactly two roots t
1 and t
2 such that t
1
<
t
0
<
t
2, t
1
u
∈
N
+, and t
2
u
∈
N
-. Since ϕ
′
t
=
A
t
-
B
t
2
p
-
1
-
C, we know that l
i
m
t
→
0
+
ϕ
′
(
t
)
=
-
C
<
0 and ϕ
′
′
t
>
0
(
∀
t
∈
(
0
,
t
0
)
). Furthermore, it follows from ϕ
′
t
1
=
0 that ϕ
(
t
1
)
<
l
i
m
t
→
0
+
ϕ
(
t
1
)
=
0. Hence, we infer from ϕ
(
t
1
)
=
J
(
t
1
u
)
≥
c
+ that c
+
<
0. Finally, we will prove that there exists ɛ
2
>
0 such that c
-
>
0 for |
g
|
p
/
p
-
1
<
ɛ
2. Since A
=
1 and B
(
u
) has upper bound, it follows from (18) that(29)
ϕ
t
0
=
t
0
2
2
-
B
2
p
t
0
2
p
-
C
t
0
=
1
2
1
2
p
-
1
B
2
/
2
p
-
2
-
B
2
p
1
2
p
-
1
B
2
p
/
2
p
-
2
-
C
1
2
p
-
1
B
1
/
2
p
-
2
=
2
p
+
1
p
-
1
2
p
2
p
-
1
2
p
/
2
p
-
2
B
-
2
/
2
p
-
2
-
C
1
2
p
-
1
B
1
/
2
p
-
2
≥
2
p
+
1
p
-
1
2
p
2
p
-
1
2
p
/
2
p
-
2
1
sup
u
E
=
1
⁡
B
u
2
/
2
p
-
2
-
C
1
2
p
-
1
sup
u
E
=
1
⁡
B
u
1
/
2
p
-
2
.Moreover, we infer from (23) and (29) that if(30)
g
p
/
p
-
1
≤
2
p
+
1
p
-
1
2
p
2
p
-
1
2
p
/
2
p
-
2
2
p
-
1
1
/
2
p
-
2
2
S
p
1
2
p
-
1
sup
u
E
=
1
⁡
B
u
1
/
2
p
-
2
=
ɛ
2
,then(31)
ϕ
t
0
≥
1
2
2
p
+
1
p
-
1
2
p
2
p
-
1
2
p
/
2
p
-
2
1
2
p
-
1
sup
u
E
=
1
⁡
B
u
2
/
2
p
-
2
≔
A
0
>
0
,where A
0 is independent of u. Hence, as in the proof of Lemma 4 we know that(32)
c
-
=
inf
u
∈
E
∖
0
⁡
max
t
≥
t
0
⁡
J
t
u
≥
ϕ
t
0
≥
A
0
>
0
.


## 3. Proof of Theorem1

In this section we are going to give the proof of the main results. Before doing this we should study the properties for the minimizing sequences for the functionalJ. In the whole paper, we say l
i
m
n
→
∞
J
′
u
n
=
0 means that l
i
m
n
→
∞
J
′
(
u
n
)
=
0.Lemma 6.
Under the assumptions of Theorem1, there exists a sequence {
u
n
}
⊂
N
+ such that J
(
u
n
)
→
c
+ and J
′
u
n
→
0 as n
→
∞.Proof.
It is easy to check thatJ is coercive and bounded from below on N
+ or N
-. So, by applying Ekeland’s variational principle (see [26]) on N
+
∪
{
0
}, one obtains a sequence u
n
⊂
N
+
∪
{
0
} satisfying(33)
1
J
u
n
≤
inf
w
∈
N
+
∪
0
⁡
J
w
+
1
n
,
(34)
2
J
u
≥
J
u
n
-
1
n
u
n
-
u
E
,
∀
u
∈
N
+
∪
0
.From Lemma 5 we know that c
+
<
0. Moreover, J
(
0
)
=
0. So we can assume that u
n
⊂
N
+ satisfying (33) and (34). Hence, in the following it suffices to prove that J
′
u
n
→
0 as n
→
∞.
In order to prove this result, we first claim that{
u
n
} is bounded. Since {
u
n
}
⊂
N
+, it follows that(35)
u
n
E
2
=
∫
R
N
∫
R
N
u
n
x
p
u
n
y
p
x
-
y
α
d
x
d
y
+
∫
R
N
g
u
n
.Furthermore, we infer from (33) that(36)
c
+
+
1
n
≥
J
u
n
=
1
2
-
1
2
p
u
n
E
2
-
1
-
1
2
p
∫
R
N
g
u
n
≥
1
2
-
1
2
p
u
n
E
2
-
1
-
1
2
p
g
p
/
p
-
1
u
n
E
,where p is given in (10). So we know that {
u
n
} is bounded. Next we claim that i
n
f
n
u
n
E
≥
σ
>
0, where σ is a constant. In fact, if not, then J
(
u
n
) would converge to zero. So we have that(37)
σ
≤
u
n
E
≤
δ
.
For anyϕ
∈
E with ϕ
≤
1, we define(38)
G
n
t
,
ω
=
J
′
u
n
+
t
ϕ
+
ω
2
u
n
u
n
+
t
ϕ
+
ω
2
u
n
.Clearly, G
n
(
0,0
)
=
0 and G
n
∈
C
1
(
R
×
R
,
R
). Moreover, one infers from {
u
n
}
⊂
N
+ that(39)
∂
G
n
∂
ω
0,0
=
1
2
Φ
′
u
n
,
u
n
>
0
.By the implicit function theorem, there exists a C
1 function ω
n
(
t
) defined on some interval (
-
τ
n
,
τ
n
), where τ
n
>
0, such that ω
n
(
0
)
=
0 and(40)
G
n
t
,
ω
n
t
=
0
,
t
∈
-
τ
n
,
τ
n
.Differentiating (40) in t at t
=
0, we obtain(41)
∂
G
n
∂
t
0,0
+
∂
G
n
∂
ω
0,0
ω
n
′
0
=
0
.We infer from {
u
n
}
⊂
N
+ and (37) that(42)
∂
G
n
∂
ω
0,0
=
1
2
Φ
′
u
n
,
u
n
=
1
2
2
u
n
E
2
-
2
p
∫
R
N
∫
R
N
u
n
x
p
u
n
y
p
x
-
y
α
d
x
d
y
-
∫
R
N
g
u
n
=
1
2
2
p
-
2
u
n
E
2
-
2
p
-
1
∫
R
N
g
u
n
≥
1
2
2
p
-
2
σ
2
-
2
p
-
1
g
p
/
p
-
1
δ
.Let ɛ
3
=
(
2
p
-
2
)
σ
2
/
2
(
2
p
-
1
)
δ. Then we know that for g
p
/
p
-
1
≤
ɛ
3
(43)
∂
G
n
∂
ω
0,0
≥
c
>
0
.Moreover, it follows from the boundedness of {
u
n
} and ϕ that(44)
∂
G
n
∂
t
0,0
=
2
Φ
′
u
n
,
ϕ
≤
c
.Hence we infer from (41)–(44) that(45)
ω
n
′
0
≤
c
.Let(46)
ϕ
¯
n
,
t
=
t
ϕ
+
ω
n
t
2
u
n
,
ϕ
n
,
t
=
u
n
+
ϕ
¯
n
,
t
.Then it follows from (40) that ϕ
n
,
t
∈
N for t
∈
(
-
τ
n
,
τ
n
). Moreover, since {
u
n
}
⊂
N
+ and N
-
∪
{
0
} is a closed set, it follows that d
i
s
t
(
u
n
,
N
-
∪
{
0
}
)
>
0. Thus, there exists 0
<
ϵ
n
<
τ
n such that for t
∈
(
-
ϵ
n
,
ϵ
n
)
(47)
ϕ
n
,
t
-
u
n
=
ϕ
¯
n
,
t
<
1
2
dist
⁡
u
n
,
N
-
∪
0
.So we know that ϕ
n
,
t
∈
N
+ for t
∈
(
-
ϵ
n
,
ϵ
n
). We deduce from (34) that(48)
J
ϕ
n
,
t
-
J
u
n
≥
-
1
n
ϕ
¯
n
,
t
E
.From Taylor theorem, we have(49)
J
ϕ
n
,
t
-
J
u
n
=
J
′
u
n
,
ϕ
¯
n
,
t
+
K
n
,
t
=
t
J
′
u
n
,
ϕ
+
K
n
,
t
,where K
(
n
,
t
)
=
o
(
∥
ϕ
¯
n
,
t
∥
E
) as t
→
0. Furthermore, it follows from (45) and (46) that(50)
lim
sup
t
→
0
⁡
ϕ
¯
n
,
t
t
E
=
lim
sup
t
→
0
⁡
ϕ
+
w
n
t
2
t
u
n
E
≤
c
,where c is independent of n. Thus, K
(
n
,
t
)
=
o
(
t
) as t
→
0. One can deduce from (48)–(50) that for any ϕ
∈
E
(51)
J
′
u
n
,
ϕ
≤
c
n
.Hence, one has that(52)
lim
n
→
∞
⁡
J
′
u
n
=
lim
n
→
∞
⁡
sup
ϕ
∈
E
,
ϕ
=
1
⁡
J
′
u
n
,
ϕ
=
0
.We finish the proof of the lemma.Next we will prove the following compactness lemma for the functionalJ.Lemma 7.
Assume thatg
∈
L
p
/
p
-
1
(
R
N
). Then J satisfies the (
P
S
)
c-condition. That is, if {
u
n
}
⊂
E such that J
(
u
n
)
→
c and J
′
u
n
→
0, as n
→
∞, then {
u
n
} has a convergent subsequence.Proof.
Let{
u
n
}
⊂
E such that J
(
u
n
)
→
c and J
′
u
n
→
0, as n
→
∞. As in Lemma 6, it is easy to check that {
u
n
} is bounded. Without loss of generality we assume that u
n
⇀
u in E and u
n
→
u in L
t
(
R
N
)
∀
t
∈
(
2
,
2
∗
). From the weak continuity of the derivative of J, we know that J
′
u
=
0. First, we have(53)
o
1
=
J
′
u
n
,
u
n
-
u
=
∫
R
N
∇
u
n
∇
u
n
-
u
+
V
x
u
n
u
n
-
u
-
∫
R
N
g
u
n
-
u
-
∫
R
N
∫
R
N
u
n
x
p
-
2
u
n
x
u
n
x
-
u
x
u
n
y
p
x
-
y
α
d
x
d
y
.We infer from Hardy-Littlewood-Sobolev inequality (see Lemma 3) that(54)
∫
R
N
∫
R
N
u
n
x
p
-
2
u
n
x
u
n
x
-
u
x
u
n
y
p
x
-
y
α
d
x
d
y
≤
c
u
n
p
-
1
u
n
-
u
s
u
n
p
s
≤
c
u
n
p
s
p
-
1
u
n
-
u
p
s
u
n
p
s
p
⟶
0
,
as
n
⟶
∞
,where s
=
2
N
/
2
N
-
α. Moreover, one can check that(55)
∫
R
N
g
u
n
-
u
⟶
0
,
as
n
⟶
∞
.Hence, it follows from (53)–(55) that(56)
∫
R
N
∇
u
n
∇
u
n
-
u
+
V
x
u
n
u
n
-
u
=
o
1
.On the other hand, it follows from J
′
u
=
0 that(57)
0
=
J
′
u
,
u
n
-
u
=
∫
R
N
∇
u
∇
u
n
-
u
+
V
x
u
u
n
-
u
-
∫
R
N
g
u
n
-
u
-
∫
R
N
∫
R
N
u
x
p
-
2
u
x
u
n
x
-
u
x
u
y
p
x
-
y
α
d
x
d
y
.Similar to (54) and (55) one has that(58)
∫
R
N
∇
u
∇
u
n
-
u
+
V
x
u
u
n
-
u
=
o
1
.So we infer from (56) and (58) that l
i
m
n
→
∞
u
n
-
u
E
=
0.Proof of Theorem1.
We first prove thatc
+ is attained by some positive u
∈
E. First, from Lemma 6, we know that there exists a sequence {
u
n
}
⊂
N
+ such that J
(
u
n
)
→
c
+ and J
′
u
n
→
0 as n
→
∞. By Lemma 7 we infer that u
n
→
u in E. Thus, J
(
u
)
=
c
+ and J
′
u
=
0. u is a nontrivial solution of (1).
Next we will prove thatc
+ is attained by positive u
1 if g is positive. As in proof (i) of Lemma 4, we know that there exists a unique t
3
>
0 such that t
3
u
∈
N
+. We claim that t
3
≥
1. In fact, since A
(
u
)
=
A
(
u
), B
(
u
)
=
B
(
u
), and C
(
u
)
≥
C
(
u
), it follows from u
∈
N
+ and proof (i) of Lemma 4 that 1
≤
t
3
<
t
0, where A
,
B
,
C, and t
0 are defined in (26) and (18). Since t
3
u
∈
N
+ is the minimum point of ϕ
t
=
J
(
t
u
) for 0
<
t
<
t
0, it follows from t
3
≥
1 that(59)
J
t
3
u
≤
J
u
.Hence, we have(60)
c
+
≤
J
t
3
u
≤
J
u
≤
J
u
=
c
+
.By Lemma 5, we know that t
3
|
u
| is nonnegative solution of (1). Furthermore, by maximum principal we infer that u
1
=
t
3
|
u
| is a positive solution of (1).
Finally we will prove that there exists positiveu
∈
N
- such that c
-
=
J
(
u
). Similar to Lemma 6, we can check that there exists a sequence {
u
n
}
⊂
N
- such that(61)
J
u
n
⟶
c
-
,
J
′
u
n
⟶
0
,
as
n
⟶
∞
.We infer from Lemma 7 that u
n
→
u in E. Thus, J
(
u
)
=
c
-
>
0 and J
′
u
=
0. u is a nontrivial solution of (1). As in proof (i) of Lemma 4, one can check that there exists unique t
4
>
0 such that t
4
|
u
|
∈
N
-. Moreover, we deduce from A
u
=
A
u, B
(
u
)
=
B
u, and C
u
≥
C
(
u
) and u
∈
N
- that t
4
≤
1. Thus, one infers from u
∈
N
- and u which is the unique maximum point of ϕ
(
t
)
=
J
(
t
u
)
∀
t
>
t
0 that(62)
c
-
≤
J
t
3
u
=
t
4
2
2
A
u
-
t
4
2
p
2
p
B
u
-
t
4
C
u
≤
t
4
2
2
A
u
-
t
4
2
p
2
p
B
u
-
t
4
C
u
=
J
t
4
u
≤
J
u
=
c
-
.So by Lemma 5, we know that t
4
|
u
| is nonnegative solution of (1). Furthermore, by maximum principal we infer that u
1
=
t
4
|
u
| is a positive solution of (1).


## 4. Solutions for the Choquard Equation with General Nonlinearity

In this section we will look for the positive solutions for the following Choquard equation with general nonlinearity:(63)
-
Δ
u
+
V
x
u
=
1
x
α
∗
F
u
f
u
+
g
x
,
u
∈
H
1
R
N
,where F
(
u
)
=
∫
0
u
‍
f
(
s
)
d
s. Since we only care about the existence of positive solutions, in what follows, we assume that f
∈
C
1
(
R
+
,
R
) verifies the following conditions.(F
1)
There existC
0
>
0 and 2
-
(
α
/
N
)
<
q
≤
p
<
(
2
N
-
α
)
/
(
N
-
2
) such that (64)
0
<
f
′
s
≤
C
0
s
q
-
1
+
s
p
-
1
.
(F
2)
There existsμ
>
2 such that (65)
0
<
μ
F
s
≤
2
f
s
s
,
∀
s
>
0
.
(F
3)
s
↦
f
′
s
/
s is strictly increasing on (
0
,
+
∞
).Then, we have the following main results.Theorem 8.
Assume that(
V
0
) and (
F
1
)
–
(
F
3
) hold. Then there exists ɛ
∗
∗
>
0 small enough such that, for every positive function g
∈
L
p
/
p
-
1
(
R
N
) with |
g
|
p
/
p
-
1
≤
ɛ
∗
∗, problem (63) admits two positive solutions. Furthermore, we know that one of the two positive solutions is a positive ground state solution and another is positive bound state solution.Since the idea of the proof of Theorem8 is the same as Theorem 1, we only state the differences. Set(66)
A
u
=
u
E
2
,
B
u
=
∫
R
N
∫
R
N
F
u
x
F
u
y
x
-
y
α
d
x
d
y
,
C
u
=
∫
R
N
g
u
.We define the functional(67)
J
u
=
1
2
A
u
-
1
2
B
u
-
C
u
,
for
u
∈
E
.It is easy to check that J
∈
C
2
(
R
,
R
), and the critical points of J are solutions of (63). Let(68)
φ
t
=
t
2
2
A
u
-
1
2
B
t
u
-
t
C
u
.Then(69)
φ
′
t
=
t
A
u
-
1
2
B
′
t
u
u
-
C
u
=
h
t
-
C
u
,where h
(
t
)
=
t
A
(
u
)
-
1
/
2
B
′
t
u
u. We claim that h
(
t
) has a unique maximum point for t
>
0. Indeed, since(70)
B
′
t
u
u
=
2
∫
R
N
∫
R
N
F
t
u
x
f
t
u
y
u
y
x
-
y
α
d
x
d
y
,we infer from condition (
F
1
) that h
(
t
)
>
0 for t
>
0 small. Moreover, one deduces from (
F
2
) that l
i
m
t
→
∞
h
(
t
)
=
-
∞. So h
(
t
) has at least one maximum point. If t
1
>
t
2 such that t
1 and t
2 are maximum points of h
(
t
), then one sees that(71)
A
u
-
1
2
B
′
′
t
1
u
u
,
u
=
0
,
A
u
-
1
2
B
′
′
t
2
u
u
,
u
=
0
.Hence we have that(72)
0
=
1
4
B
′
′
t
1
u
u
,
u
-
B
′
′
t
2
u
u
,
u
=
∫
R
N
∫
R
N
f
t
1
u
x
u
x
f
t
1
u
y
u
y
x
-
y
α
d
x
d
y
︸
l
1
+
∫
R
N
∫
R
N
F
t
1
u
x
f
′
t
1
u
y
u
2
y
x
-
y
α
d
x
d
y
︸
l
2
-
∫
R
N
∫
R
N
f
t
2
u
x
u
x
f
t
2
u
y
u
y
x
-
y
α
d
x
d
y
︸
l
3
-
∫
R
N
∫
R
N
F
t
2
u
x
f
′
t
2
u
y
u
2
y
x
-
y
α
d
x
d
y
︸
l
4
.Obviously, one has(73)
l
1
-
l
3
=
∫
R
N
∫
R
N
f
t
1
u
x
-
f
t
2
u
x
u
x
f
t
1
u
y
u
y
x
-
y
α
d
x
d
y
+
∫
R
N
∫
R
N
f
t
2
u
y
u
y
f
t
1
u
y
-
f
t
2
u
y
u
y
x
-
y
α
d
x
d
y
.Moreover, it follows from (
F
3
) that(74)
f
t
1
u
-
f
t
2
u
=
∫
0
t
1
u
f
′
s
d
s
-
∫
0
t
2
u
f
′
s
d
s
=
∫
t
2
u
t
1
u
f
′
s
d
s
=
∫
t
2
u
t
1
u
f
′
s
s
s
d
s
≥
f
′
t
1
u
t
1
-
t
2
u
>
0
.So we infer from (73) and (74) that(75)
l
1
-
l
3
>
0
.Similar to (73) and (74) one infers from (
F
3
) that(76)
l
2
-
l
4
=
∫
R
N
∫
R
N
F
t
1
u
x
-
F
t
2
u
x
f
′
t
1
u
y
u
y
x
-
y
α
d
x
d
y
+
∫
R
N
∫
R
N
F
t
2
u
x
f
′
t
1
u
y
u
y
-
f
′
t
2
u
y
u
y
x
-
y
α
d
x
d
y
>
0
.This contradicts with (72). So h
(
t
) has a unique maximum point. By using this property of h, one can establish Lemmas 4 and 5 for |
g
|
p
/
p
-
1 sufficiently small.In the proof of Lemma6, we need check the following differences: (i) the boundedness of {
u
n
}
⊂
N
+; (ii) ∂
G
n
/
∂
ω
(
0,0
)
≥
σ
0
>
0, where G
n
(
t
,
ω
)
=
J
′
(
u
n
+
t
ϕ
+
ω
/
2
u
n
)
u
n
+
t
ϕ
+
ω
/
2
u
n. We first check the boundedness of {
u
n
}
⊂
N
+. Similar to (36) one infers from {
u
n
}
⊂
N
+ and (
F
2
) that(77)
c
+
+
1
n
≥
J
u
n
=
J
u
n
-
1
μ
J
′
u
n
u
n
=
1
2
μ
∫
R
N
∫
R
N
F
u
n
x
2
f
u
n
y
u
n
y
-
μ
F
u
n
y
x
-
y
α
+
1
2
-
1
μ
u
n
E
2
-
1
-
1
μ
∫
R
N
g
u
n
≥
1
2
-
1
μ
u
n
E
2
-
1
-
1
μ
∫
R
N
g
u
n
.Therefore, {
u
n
} is bounded.On the other hand, we prove that∂
G
n
/
∂
ω
(
0,0
)
≥
σ
0
>
0. Similar to (42) one deduces from {
u
n
}
⊂
N
+ that(78)
2
∂
G
n
∂
ω
0,0
=
2
u
n
E
2
-
∫
R
N
∫
R
N
F
u
n
x
f
u
n
y
u
n
y
x
-
y
α
-
∫
R
N
∫
R
N
f
u
n
x
u
n
x
f
u
n
y
u
n
y
x
-
y
α
-
∫
R
N
∫
R
N
F
u
n
x
f
′
u
n
y
u
n
2
y
x
-
y
α
-
∫
R
N
g
u
n
=
u
n
E
2
-
∫
R
N
∫
R
N
f
u
n
x
u
n
x
f
u
n
y
u
n
y
x
-
y
α
-
∫
R
N
∫
R
N
F
u
n
x
f
′
u
n
y
u
n
2
y
x
-
y
α
.Arguing by contradiction, if 2
∂
G
n
/
∂
ω
(
0,0
)
→
0 as n
→
∞, we infer from {
u
n
}
⊂
N
+ that(79)
∫
R
N
∫
R
N
f
u
n
u
n
-
F
u
n
x
f
u
n
y
u
n
y
x
-
y
α
=
∫
R
N
g
u
n
-
∫
R
N
∫
R
N
F
u
n
x
f
′
u
n
y
u
n
2
y
x
-
y
α
.We conclude from (
F
1
) and (
F
2
) that(80)
∫
R
N
∫
R
N
F
u
n
x
f
′
u
n
y
u
n
2
y
x
-
y
α
>
0
,
∫
R
N
∫
R
N
f
u
n
u
n
-
F
u
n
x
f
u
n
y
u
n
y
x
-
y
α
≥
μ
2
-
1
∫
R
N
∫
R
N
F
u
n
x
f
u
n
y
u
n
y
x
-
y
α
.Moreover, since u
n
E
≥
δ
0
>
0, it follows from {
u
n
}
⊂
N
+ that if |
g
|
p
/
p
-
1 is sufficiently small,(81)
∫
R
N
∫
R
N
F
u
n
x
f
u
n
y
u
n
y
x
-
y
α
≥
c
0
>
0
.So substituting (80) and (81) into (79), we obtain a contradiction if |
g
|
p
/
p
-
1 is sufficiently small. Hence, the conclusions of Lemma 6 hold. Thus, following the line of proof of Theorem 1, one can prove the results of Theorem 8.Remark 9.
Our arguments in this paper can also apply to the following generalized nonlinear Choquard equation with perturbation:(82)
-
Δ
u
+
V
x
u
=
K
α
x
∗
u
p
u
p
-
2
u
+
g
x
,
x
∈
R
N
,where the dimension N
∈
N
=
{
1,2
,
…
} is given, V
∈
C
(
R
N
,
R
+
) is an external potential, and K
α
:
R
N
∖
{
0
}
→
R is the Riesz potential [27] of order α
∈
(
0
,
N
) defined for every x
∈
R
N
∖
{
0
} by(83)
K
α
x
=
Γ
N
-
α
/
2
Γ
α
/
2
π
N
/
2
2
α
x
N
-
α
,
p
∈
[
2
,
N
+
α
/
N
-
2
+
). The results of Theorem 1 remain true for (82).



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*Source: 760157-2015-09-30.xml*