Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 75, pp. 1–20.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu

DOI: 10.58997/ejde.2023.75

CONCENTRATION OF NODAL SOLUTIONS FOR

SEMICLASSICAL QUADRATIC CHOQUARD EQUATIONS

LU YANG, XIANGQING LIU, JIANWEN ZHOU

Abstract. In this article concerns the semiclassical Choquard equation−ε2∆u+

V (x)u = ε−2( 1
|·| ∗ u

2)u for x ∈ R3 and small ε. We establish the existence of

a sequence of localized nodal solutions concentrating near a given local mini-

mum point of the potential function V , by means of the perturbation method
and the method of invariant sets of descending flow.

1. Introduction

In the past two decades, attention has devoted to the study of the existence,
multiplicity, and properties of the solutions for the nonlinear Choquard equation

− ε2∆u+ V (x)u = εα−N (
1

| · |α
∗ up)up−1, x ∈ RN , (1.1)

where 0 < α < N , 2N−α
N < p < 2N−α

N−2 , and ε > 0 is a small positive parameter.
When N = 3, α = 1 and ε = 1, as an important model, the problem

∆u+ V (x)u =
( 1

| · |
∗ u2

)
u, x ∈ R3 (1.2)

was introduced by Pekar [30] to describe the quantum theory of a polaron at rest,
and then used by Choquard [18] to study steady states of the one-component plasma
approximation to the Hartree-Fock theory. Later, the same equation re-emerged as
a model of self-gravitating matter [29], and in that context it is referred as to the
Schrödinger-Newton system.

For the existence and qualitative properties of solutions for the nonlinear Choquard
equation (1.1), we refer the reader to [2, 3, 5, 9, 10, 14, 17, 26, 27, 31, 32, 34, 35]
and references therein. In particular, for p > 2, the existence of nodal solu-
tions for the Choquard equation is an appealing aspect which is investigated in
[5, 8, 11, 13, 15, 16, 23] by the variational method. In the physical case, for p = 2,
the existence of nodal solutions for (1.1) only has few results. For p ≥ 2 and V
is a radial symmetry function, Gui [13] show that, for any positive integer k, the
equation(1.1) has a sign-changing solution uk which changes signs exactly k times.
When V ≡ 1 and p = 2, Ghimenti[12] proved the existence of the least action

2020 Mathematics Subject Classification. 35B05, 35B45.
Key words and phrases. Choquard equation; sign-changing solution; perturbation method;

Fredholm operator; nodal solution.
©2023. This work is licensed under a CC BY 4.0 license.

Submitted May 18, 2023. Published October 30, 2023.

1



2 L. YANG, X. LIU, J. ZHOU EJDE-2023/75

nodal solutions. However, without symmetry or periodicity assumptions on the
potential function V , there is no result of the existence of infinitely many sign-
changing solutions for the equation (1.1) with p = 2. Motivated by the works
mentioned above, we consider the existence of infinitely sign-changing solutions for
the following equation

− ε2∆u+ V (x)u = ε−2(
1

| · |
∗ u2)u, x ∈ R3, (1.3)

where the potential function V satisfies the assumptions:

(A1) V ∈ C1(R3,R) and there exist constants b > a > 0 such that

a ≤ V (x) ≤ b, ∀x ∈ R3 .

(A2) There exists a bounded domain M⊂ R3 with smooth boundary ∂M such
that

〈−→n (x), ∇V (x)〉 > 0 , ∀x ∈ ∂M ,

where −→n (x) is the outer normal of ∂M at x.

Under the assumption (A2), in view of the critical set

A = {x ∈M : ∇V (x) = 0} 6= ∅, (1.4)

without loss of generality we assume 0 ∈ A. For each set B ⊂ RN and any δ > 0,
we set

Bδ = {x ∈ R3 : δx ∈ B},

Bδ = {x ∈ R3 : dist(x,B) := inf
y∈B
|x− y| < δ}.

The main result of this paper reads as follows.

Theorem 1.1. Assume that (A1), (A2) hold. For each positive integer k, there
exists ε′k > 0 such that if 0 < ε < ε′k, then (1.3) has at least k pairs of sign-
changing solutions ±vj,ε, j = 1, . . . , k. Moreover, for any δ > 0 there exist µ > 0,
C = Ck > 0, and ε′k(δ) > 0 such that if 0 < ε < ε′k(δ), then

|vj,ε(x)| ≤ Ce−
µ
ε dist(x,Aδ) for x ∈ R3, j = 1, . . . , k. (1.5)

In this article, we can also obtain the existence and concentration phenomenon
of the sign-changing solution of the equation(1.1). Here, we only consider the case
with α = 1 and N = 3.

By making the change of variable εy = x, equation (1.3) is equivalent to

−∆u+ V (εx)u =
( 1

| · |
∗ u2

)
u, x ∈ R3 (1.6)

and the corresponding functional is

Iε(u) =
1

2

∫
R3

(|∇u|2 + V (εx)u2) dx− 1

4

∫
R3

(
1

| · |
∗ u2)u2 dx . (1.7)

We will use the method of invariant sets of descending flow to prove the existence
of sign-changing solutions for (1.3), but the setting of invariant sets of descending
flow can not fit well for the Choquard equation. In [15], we used the perturbation
method [24] to overcome this difficulty for Choquard equation (1.1) with 2 < p <
2N−α
N−2 . However, the method described in [15] becomes invalid for the case p = 2.

To obtain compactness for the functional Iε, we use the penalization method
in [4, 36]. Let G ∈ C∞(R,R), satisfy G′(s) ∈ [0, 1], G′′(s) ∈ [0, 2], G(s) = 0 for



EJDE-2023/75 QUADRATIC CHOQUARD EQUATIONS 3

s ≤ 1/2 and G(s) = s− 1 for s ≥ 3/2. We also require that |G(s)−G′(s)s| ≤ 3/2.
We define

χε(x) =

{
0, if x ∈Mε

ε−6ζ(dist(x,Mε)), if x /∈Mε,

where ζ ∈ C∞ is a cut-off function such that ζ(t) = 0 if t ≤ 0; ζ(t) = 1 if t ≥ 1
and 0 ≤ ζ ′(t) ≤ 2, 0 ≤ ζ(t) ≤ 1. For each ε > 0, p ∈ (2, p0), p0 ∈ (2, 5) is a fixed
constant, u ∈ H1(R3), we consider functionals:

Γε(u) =
1

2

∫
R3

(|∇u|2 + V (εx)u2) dx+
1

2
G
(∫

R3

χε(x)u2 dx
)

− 1

4

∫
R3

( 1

| · |
∗ u2

)
u2 dx, u ∈ H1(R3),

(1.8)

and

Γε,p(u) =
1

2

∫
R3

(|∇u|2 + V (εx)u2) dx+
1

2
G
(∫

R3

χε(x)u2 dx
)

− 1

2p

∫
R3

( 1

| · |
∗ |u|p

)
|u|p dx, u ∈ H1(R3).

(1.9)

Note that

〈DΓε(u), ϕ〉

=

∫
R3

(∇u∇ϕ+ V (εx)uϕ) dx+G′
(∫

R3

χε(x)u2 dx
)∫

R3

χε(x)uϕdx

−
∫
R3

( 1

| · |
∗ u2

)
uϕdx, ∀ϕ ∈ H1(R3),

(1.10)

and

〈DΓε,p(u), ϕ〉

=

∫
R3

(∇u∇ϕ+ V (εx)uϕ) dx+G′
(∫

R3

χε(x)u2 dx
)∫

R3

χε(x)uϕdx

−
∫
R3

( 1

| · |
∗ |u|p

∣∣u|p−2uϕdx, ∀ϕ ∈ H1(R3).

(1.11)

We also note that the critical points of Γε and Γε,p are, respectively, solutions of

−∆u+ V (εx)u+G′
(∫

R3

χε(x)u2 dx
)
χε(x)u =

( 1

| · |
∗ u2

)
u, (1.12)

−∆u+ V (εx)u+G′(

∫
R3

χε(x)u2 dx)χε(x)u =
( 1

| · |
∗ |u|p

)
|u|p−2u, (1.13)

for all u ∈ H1(R3). If u is a critical point of Γε and
∫
R3 χε(x)u2 dx < 1

2 , then u is
a solution of (1.6).

Let b ∈ C∞(R+, [0, 1]) such that b(t) = 1 if t ≤ 1; b(t) = 0 if t ≥ 2 and

0 ≤ b(t) ≤ 1, b′(t) ≤ 0. Let 0 < λ < 1, bλ(t) = b(λt),mλ(t) =
∫ t

0
bλ(τ) dτ ,

gλ(t) = mλ(t)
t . We define

ψ(u) =

∫
R3

( 1

| · |
∗ |u|p

)
|u|p dx



4 L. YANG, X. LIU, J. ZHOU EJDE-2023/75

and

Γ(λ)
ε,p (u) =

1

2

∫
R3

(|∇u|2 + V (εx)u2) dx+
1

2
G
(∫

R3

χε(x)u2 dx
)

− 1

2p
gλ(ψ1/2(u))ψ(u).

(1.14)

For each ϕ ∈ H1(R3), since g′λ(t)t+ gλ(t) = bλ(t), we have

〈DΓ(λ)
ε,p (u), ϕ〉

=

∫
R3

∇u∇ϕ+ V (εx)uϕdx+G′
(∫

R3

χε(x)u2 dx
)∫

R3

χε(x)uϕdx

− 1

2
(bλ(ψ1/2(u)) + gλ(ψ1/2(u)))

∫
R3

( 1

| · |
∗ |u|p

)
|u|p−2uϕdx.

(1.15)

Define ‖u‖2 =
∫
R3(|∇u|2 + u2) dx for u ∈ H1(R3). By Hardy-Littlewood-Sobolev

inequality we know that there exists Cp > 0 such that ψ1/2(u) ≤ Cp‖u‖p and Cp

independent of u. It is easy to know that when ‖u‖ ≤ ( 1
Cpλ

)
1
p , we have Γε,p(u) =

Γ
(λ)
ε,p (u) and DΓε,p(u) = DΓ

(λ)
ε,p (u).

This article is organized as follows. In Section 2, we prove (PS)c condition for
Γε,p and give some uniform estimates (independent of p) on the critical points of
Γε,p. In Section 3, we prove the existence of sign-changing solutions for Γε. Section
4 is devoted to the proof of Theorem 1.1.

For a Banach space E, we denote its dual space by E′. Throughout the paper,
c, c0, c1, . . . denote different constants and cλ, Cλ denote constants depending on λ.

2. (PS)c condition for Γε,p

In this section, we first collect elementary properties of the Choquard term, and
then we prove that Γε,p satisfies the (PS)c condition.

Lemma 2.1 (Hardy-Littlewood-Sobolev inequality [19]). Suppose α ∈ (0, N), and
s, r > 1 with 1

s + 1
r + α

N = 2. Let g ∈ Ls(RN ), h ∈ Lr(RN ), there exists a sharp
constant C(s, α, r,N), independently of g, h, such that∫

RN

∫
RN

g(x)h(y)

|x− y|α
dx dy ≤ C‖g‖Lr(RN )‖h‖Ls(RN ).

Lemma 2.2. Assume un ⇀ u in H1(R3), then∫
R3

( 1

| · |
∗ |un|p)|un|p dx−

∫
R3

(
1

| · |
∗ |un − u|p)|un − u|p dx→

∫
R3

(
1

| · |
∗ |u|p)|u|p dx

and

(
1

| · |
∗ |un|p)|un|p−2un − (

1

| · |
∗ |un − u|p)|un − u|p−2(un − u)→ (

1

| · |
∗ |u|p)|u|p−2u

in (H1(R3))′.

The above lemma can be proved as in [1, Lemma 3.4].

Lemma 2.3. It holds that for t > 0 and 0 < λ < 1:

(1) gλ(t) = 1, g′λ(t) = 0 if 0 < t < 1/λ;
(2) −g′λ(t)t ≤ gλ(t) ≤ cλ/t, where cλ =

∫∞
0
b(τ) dτ/λ;

(3) bλ(t)t ≤ gλ(t)t ≤ cλ.



EJDE-2023/75 QUADRATIC CHOQUARD EQUATIONS 5

The above lemma can be obtained by direct calculation.

Lemma 2.4. Assume ‖DΓ
(λ)
ε,p (u)‖ ≤ 1,Γ

(λ)
ε,p (u) ≤ L, then there exists λL,p > 0

such that if 0 < λ < λL,p, then DΓ
(λ)
ε,p (u) = DΓε,p(u),Γ

(λ)
ε,p (u) = Γε,p(u).

Proof. By (1.14),(1.15) and Lemma 2.3, we have

L+ ‖u‖ ≥ Γ(λ)
ε,p (u)− 1

p
〈DΓ(λ)

ε,p (u), u〉

= (
1

2
− 1

p
)

∫
R3

(|∇u|2 + V (εx)u2) dx+
1

2
G
(∫

R3

χε(x)u2 dx
)

− 1

p
G′
(∫

R3

χε(x)u2 dx
)∫

R3

χε(x)u2 dx

+
1

2p
bλ(ψ1/2(u))

∫
R3

(
1

| · |
∗ |u|p)|u|p dx

≥ cp
∫
R3

(|∇u|2 + V (εx)u2) dx+ cpG
(∫

R3

χε(x)u2 dx
)
− c .

(2.1)

As a result, there exists a constant cL,p such that ψ1/2(u) ≤ c‖u‖p ≤ cL,p and
cL,p → ∞ as p → 2. Choose λL,p ≤ 1

2cL,p
, by Lemma 2.3, we have DΓλε,p(u) =

DΓε,p(u),Γλε,p(u) = Γε,p(u) . �

Lemma 2.5. For each L > 0, there exists εL > 0 such that, for 0 < ε < εL, if
c < L, the following statements hold

(1) The Γε,p satisfies the (PS)c condition.
(2) Let pn ⊂ (2, 2∗) be a sequence such that pn → 2 as n → ∞. For {un} ⊂

H1(R3) such that

Γε,pn(un)→ c,DΓε,pn(un) = on(1),

there exists a critical point u ∈ H1(R3) of Γε, such that un → u ∈ H1(R3)
up to a subsequence.

Proof. (1) The proof of this part is similar to that in [15].
(2) By (1.9) and (1.11), we have

L ≥ Γε,pn(un)− 1

2pn
〈DΓε,pn(un), un〉

= (
1

2
− 1

2pn
)

∫
R3

(|∇un|2 + V (εx)u2
n) dx+

1

2
G
(∫

R3

χε(x)u2
n dx

)
− 1

2pn
G′
(∫

R3

χε(x)u2
n dx

)∫
R3

χε(x)u2
n dx

≥c
∫
R3

(|∇un|2 + V (εx)u2
n) dx+ cG

(∫
R3

χε(x)u2
n dx

)
− c .

(2.2)

By (2.2) we know that there exists η̂L > 0 independent of ε, p such that ‖un‖ ≤ η̂L
and G(

∫
R3 χε(x)u2

n dx) ≤ η̂L. Up to a subsequence, we assume that un ⇀ u in

H1(R3),

ζn := G′
(∫

R3

χε(x)u2
n dx

)
→ ζ.



6 L. YANG, X. LIU, J. ZHOU EJDE-2023/75

It is easy to show that u is a solution of the equation

−4u+ V (εx)u+ ζχε(x)u = (
1

| · |
∗ u2)u. (2.3)

By Lemma 2.2, for any v ∈ H1(R3),

o(‖v‖) = 〈DΓε,pn(un), v〉

=

∫
R3

(∇(un − u)∇v + V (εx)(un − u)v) dx

+ ζ

∫
R3

χε(x)(un − u)v dx+ (ζn − ζ)

∫
R3

χε(x)unv dx

−
∫
R3

(
1

| · |
∗ |un − u|pn)|un − u|pn−2(un − u)v dx

−
∫
R3

(
(

1

| · |
∗ |u|pn)|u|pn−2u− (

1

| · |
∗ u2)u

)
v dx+ o(‖v‖),

(2.4)

as n → ∞. Since ‖un‖ ≤ η̂L, we have ‖u‖ ≤ η̂L. Hence, there exists a constant
such that

1

| · |
∗ |u|pn ≤ c, 1

| · |
∗ u2 ≤ c,∣∣( 1

| · |
∗ |u|pn)|u|pn−2u− (

1

| · |
∗ u2)u

∣∣2 ≤ c(|u|pn−1 + u)2 ≤ c(|u|2(p0−1) + u2).

Using the dominated convergence theorem, we obtain∫
R3

(
(

1

| · |
∗ |u|pn)|u|pn−2u− (

1

| · |
∗ u2)u

)
v dx = o(‖v‖).

Choose R0 > 0 such that M ⊂ B(0, R0). Let φε be a C∞ function such that
φε(x) = 0 for |x| ≤ ε−1(R0 +1)+1; φε(x) = 1 for |x| ≥ ε−1(R0 +1)+2; 0 ≤ φε ≤ 1
and |∇φε| ≤ 4. Take v = φ2

ε(un − u) in (2.4), we obtain∫
R3

(|∇(φε(un − u))|2 + V (εx)φ2
ε(un − u)2) dx−

∫
R3

(un − u)2|∇φε|2 dx

+ ζ

∫
R3

χε(x)φ2
ε(un − u)2 dx+ (ζn − ζ)

∫
R3

χε(x)φ2
εun(un − u) dx

−
∫
R3

( 1

| · |
∗ |un − u|pn

)
|un − u|pn−2φ2

ε(un − u)2 dx = o(1), n→∞.

(2.5)

Since ζn → ζ, n→∞ and |∇φε|2 has a compact support, by un ⇀ u in H1(R3) we
have∫

R3

(un − u)2|∇φε|2 dx = o(1), (ζn − ζ)

∫
R3

χε(x)φ2
εun(un − u) dx = o(1),



EJDE-2023/75 QUADRATIC CHOQUARD EQUATIONS 7

as n→∞. Then by (A1), ‖u‖ ≤ η̂L and (2.5), we obtain

min{1, a}‖φε(un − u)‖2

≤
∫
R3

(
1

| · |
∗ |un − u|pn)|un − u|pn−2φ2

ε(un − u)2 dx+ o(1)

≤
∫
R3

∫
|y|≤ε−1R0+1

|un(y)− u(y)|pn |un(x)− u(x)|pnφ2
ε(x)

|x− y|
dx dy

+

∫
R3

∫
|y|≥ε−1R0+1

|un(y)− u(y)|pn |un(x)− u(x)|pnφ2
ε(x)

|x− y|
dx dy + o(1)

≤ c‖un − u‖
L

6pn
5 (Bε−1R0+1(0))

‖un − u‖pn−2

L
6pn
5 (R3)

‖φε(un − u)‖2

+ c‖φε(un − u)‖2
(∫
|x|≥ε−1R0+1

|un|
6pn
5 dx

+

∫
|x|≥ε−1R0+1

|u|
6pn
5 dx

) 5(2pn−2)
6pn

+ o(1)

≤ o(1)‖φε(un − u)‖2 + c1‖φε(un − u)‖2
(∫
|x|≥ε−1R0+1

|un|
6pn
5 dx

+

∫
|x|≥ε−1R0+1

|u|
6pn
5 dx

) 5(2pn−2)
6pn

+ o(1), n→∞.

(2.6)

Since (Mε)
1 ⊂ B(0, ε−1R0 + 1) and G(

∫
R3 χε(x)u2

n dx) ≤ η̂L, we have∫
|x|≥ε−1R0+1

u2
n dx ≤ c(

1

2
+ η̂L)ε6. (2.7)

By Fatou’s Lemma, we have∫
|x|≥ε−1R0+1

u2 dx ≤ c(1

2
+ η̂L)ε6. (2.8)

Using the interpolation inequality, for 2 < 6pn
5 < q < 2∗, we have

‖un‖
L

6pn
5 (R3)

≤ ‖un‖tnL2(R3)‖un‖
1−tn
Lq(R3) ≤ c‖un‖

tn
L2(R3)‖un‖

1−tn , (2.9)

where 5
6pn

= tn
2 + (1−tn)

q , 0 < t0 < tn < 1. Combining with ‖un‖ ≤ η̂L, (2.7) and

(2.8) we obtain∫
|x|≥ε−1R0+1

|un|
6pn
5 dx ≤ CLε

18pnt0
5 ,

∫
|x|≥ε−1R0+1

|u|
6pn
5 dx ≤ CLε

18pnt0
5 , (2.10)

where CL is independent of ε. Choose εL > 0 such that c1 ·(C
5(2pn−2)

6pn

L ε
3(2pn−2)t0
k ) ≤

min{1, a}/2. Then by (2.6), for 0 < ε < εL, we have

lim
n→∞

‖φε(un − u)‖ = 0. (2.11)

Set v = (1− φε)2(un − u) in (2.4), it is easy to obtain

lim
n→∞

‖(1− φε)(un − u)‖ = 0. (2.12)

The result of the lemma follows from (2.11) and (2.12). �



8 L. YANG, X. LIU, J. ZHOU EJDE-2023/75

3. Existence of sign-changing critical points for Γε

To obtain multiple sign-changing critical points of Γλε,p, we introduce the abstract
critical point theorem [22, Theorem 2.5], see also [6, Theorem 3.2].

Let X be a Hilbert space, f be an even C2-functional on X. Let P,Q be open
convex sets of X, Q = −P. Set

W = P ∪Q, Σ = ∂P ∩ ∂Q.
For a critical point x ∈ X of f , the augmented Morse index

m∗(x) = max{dimX0 : X0 ⊂ Xis a subspace such that D2f(x)(h, h) ≤ 0,∀h ∈ X}.
Assume

(A3) there exists L > 0 such that f satisfies the (PS)c condition, for c < L;
(A4) c∗ = infx∈Σ f(x) > 0 ;
(A5) For every critical point x of f , D2f is a Fredholm operator.

Also assume there exists an odd continuous map A : X → X satisfying

(A6) given c0, b0 > 0, there exists b = b(c0, b0) > 0 such that if ‖Df(x)‖ ≥
b0, |f(x)| ≤ c0, then

〈Df(x), x−Ax〉 ≥ b‖x−Ax‖ > 0 ;

(A7) A(∂Pj) ⊂ Pj , A(∂Qj) ⊂ Qj , j = 1, . . . , k .

We define

Γj = {E|E ⊂ X, E compact, −E = E, γ(E ∩ η−1(Σ)) ≥ j for η ∈ Λ},
Λ =

{
η : η ∈ C(X,X) : η is odd, η(P ) ⊂ P, η(Q) ⊂ Q, j = 1, . . . , k,

η(x) = x if f(x) < 0
}

where γ is the genus of symmetric sets,

γ(E) = inf
{
n : there exists an odd map η : E → Rn\{0}

}
.

Assume that

(A8) Γj is nonempty.

Define

cj = inf
E∈Γj

sup
x∈E\W

f(x), j = 1, 2, . . . ,

Kc = {x : Df(x) = 0, f(x) = c}, K∗c = Kc \W .

Theorem 3.1. Assume (A3)–(A8) hold. If cj < L, j = 1, . . . , k, then

(1) cj ≥ c∗, K∗cj 6= ∅;
(2) There exists x ∈ Kcj\W with m∗(x) ≥ j.

For u ∈ H1(R3), we define v = Au by the unique solution to

−∆v + V (εx)v +G′
(∫

R3

χε(x)u2 dx
)
χε(x)v

=
1

2
(bλ(ψ1/2(u)) + gλ(ψ1/2(u)))(

1

| · |
∗ |u|p)|u|p−2u .

(3.1)

Note that A is odd, well defined, and continuous on H1(R3); see [15, Lemma 3.1].

Lemma 3.2. Let u ∈ H1(R3). If v = Au, then



EJDE-2023/75 QUADRATIC CHOQUARD EQUATIONS 9

(1) 〈DΓ
(λ)
ε,p (u), u− v〉 ≥ c‖u− v‖2;

(2) ‖DΓ
(λ)
ε,p (u)‖ ≤ c(1 + |Γ(λ)

ε,p (u)|+ ‖u− v‖)γ‖u− v‖.

This lemma can be proved as in [15, Lemma 3.2]. Using Lemma 3.2, it is easy
to prove assumption (A6).

For δ > 0, let

P := {u ∈ H1(R3) : ‖u+‖L6p/5(R3) < δ},
Q := {u ∈ H1(R3) : ‖u−‖L6p/5(R3) < δ} .

Lemma 3.3. For 0 < λ < 1, there exists δλ > 0 such that for 0 < δ < δλ,

A(∂P ) ⊂ P, A(∂Q) ⊂ Q, δλ → 0 as λ→ 0.

Moreover, there exists δ0 ∈ (0, δλ) and c∗ = c∗(δ0) > 0 such that

Γ(λ)
ε,p (u) ≥ c∗, for u ∈ ∂P ∩ ∂Q.

The proof of the above lemma is similar to that of [15, Lemmas 3.4 and 3.5].
Let

J0(u) =
1

2

∫
B(0,1)

(|∇u|2 + bu2) dx− c
(∫

B(0,1)

u2 dx
)2

,

where c independent of p, λ. Let {en} ⊂ H1
0 (B(0, 1)) be an orthogonal basis and

Hn := span{e1, . . . , en}. Then there exists an increasing sequence {Rn} such that

J0(u) < 0, ∀u ∈ Hn, ‖u‖ ≥ Rn.
Choose an appropriate ε such that

B(0, 1) ⊂Mε. (3.2)

Define ϕn ∈ C(Bn, H
1
0 (B(0, 1))) as

ϕn(t) = Rn

n∑
i=1

tiei, t = (t1, . . . , tn) ∈ Bn .

Let

Γj = {E ⊂ H1(R3) : E is compact,−E = E, γ(E ∩ η−1(Σ)) ≥ j for η ∈ Λ},
Λ =

{
η ∈ C(H1(R3) : H1(R3)), η is odd, η(P ) ⊂ P, η(Q) ⊂ Q,

η(u) = u if Γ(λ)
ε,p (u) < 0

}
.

Lemma 3.4. There exists λ̃k > 0, such that, for 0 < λ < λ̃k and sufficiently small
ε, Ej = ϕj+1(Bj+1) ⊂ Γj , j = 1, . . . , k.

Proof. For x ∈ Mε, we have χε(x) = 0 and V (εx) ≤ b. Then for u ∈ Ej , we have

G(
∫
R3 χεu

2 dx) = 0. For u ∈ Ej , choose λ̃k = c(Rk) such that, for 0 < λ < λ̃k,

gλ(ψ1/2(u)) = 1. Then we have

Γ(λ)
ε,p (u) =

1

2

∫
B(0,1)

(|∇u|2 + V (εx)u2) dx− 1

2p
ψ(u)

≤ 1

2

∫
B(0,1)

(|∇u|2 + bu2) dx− 1

4

(∫
B(0,1)

|u|p dx
)2

≤ 1

2

∫
B(0,1)

(|∇u|2 + bu2) dx− c
(∫

B(0,1)

|u|2 dx
)2



10 L. YANG, X. LIU, J. ZHOU EJDE-2023/75

≤ J0(u) .

As for [21, Lemma 5.6], we can complete the proof. �

Lemma 3.5. For every critical point u of Γε,p, D2Γε,p(u) is a Fredholm operator.

Proof. Note that every critical point of Γε,p is a weak solution of

−∆u+ V (εx)u+G′
(∫

R3

χε(x)u2 dx
)
χε(x)u = (

1

| · |
∗ |u|p)|u|p−2u,

u ∈ H1(R3).

(3.3)

Assume u solves (3.3), by the sub-solution estimates, we have u ∈ L∞(R3) and
u(x)→ 0 as x→∞. For ψ,ϕ ∈ H1(R3), we have

〈D2Γε,p(u)ψ,ϕ〉

=

∫
R3

(∇ψ∇ϕ+ V (εx)ψϕ) dx+G′
(∫

R3

χϕ(x)u2 dx
)∫

R3

χε(x)ψϕdx

+G′′
(∫

R3

χε(x)u2 dx
)∫

R3

χε(x)uψ dx

∫
R3

χε(x)uϕdx

+ (p− 1)

∫
R3

(
1

| · |
∗ |u|p)|u|p−2ψϕdx+ p

∫
R3

(
1

| · |
∗ (|u|p−2uψ))|u|p−2uϕdx

Note −∆ +V (εx) +G′(
∫
R3 χε(x)u2 dx)χε(x) : H1(R3)→ H−1(R3) is the Fredholm

operator. On the other hand, the linear operator Q : H1(R3) → H−1(R3) defined
by

〈Qψ,ϕ〉 = G′′
(∫

R3

χε(x)u2 dx
)∫

R3

χε(x)uψ dx

∫
R3

χε(x)uϕdx

+ (p− 1)

∫
R3

(
1

| · |
∗ |u|p)|u|p−2ψϕdx

+ p

∫
R3

( 1

| · |
∗ (|u|p−2uψ)

)
|u|p−2uϕdx

(3.4)

is compact. Hence D2Γε,p(u) is the Fredholm operator. �

We define

cj(ε, p, λ) = inf
E∈Γj

sup
u∈E\W

Γ(λ)
ε,p (u), c̃j = sup

u∈Ej+1

J0(u), j = 1, . . . , k.

Theorem 3.6. Let L > 0, 0 < λ < λ
(k)
L,p, 0 < ε < εL, k be such that c̃k < L. Γε,p

has at least k pairs of sign-changing critical points {±uj,ε,p, 1 ≤ j ≤ k} such that

Γε,p(uj,ε,p) = cj(ε, p) ≤ c̃k, 1 ≤ j ≤ k.
Moreover, there exists uj,ε,p ∈ K∗cj(ε,p) such that m∗(uj,ε,p) ≥ j.

Proof. By the definition of cj(ε, λ), c̃j and Lemma 3.4, for 0 < λ < λ̃k, we obtain

c1(ε, p, λ) ≤ · · · ≤ ck(ε, p, λ) ≤ c̃k, j = 1, . . . , k.

By Lemma 2.4, there exists λL,p > 0 such that if 0 < λ < λ
(k)
L,p := min{λL,p, λ̃k},

DΓ
(λ)
ε,p (u) = o(1) and Γ

(λ)
ε,p (u) ≤ L, then we have DΓ

(λ)
ε,p (u) = DΓε,p(u),Γ

(λ)
ε,p (u) =

Γε,p(u). By Lemma 2.5 and Lemma 3.5, for p ∈ (2, p0), 0 < λ < λ
(k)
L,p, Γ

(λ)
ε,p satisfies

(PS)c condition with c < L and D2Γ
(λ)
ε,p (u) is the Fredholm operator for some u



EJDE-2023/75 QUADRATIC CHOQUARD EQUATIONS 11

such that DΓ
(λ)
ε,p (u) = 0,Γ

(λ)
ε,p (u) < L. Hence, we have verified that Γ

(λ)
ε,p satisfies

all assumptions of Theorem 3.1. By Theorem 3.1, for 0 < ε < εL, we obtain

Γ
(λ)
ε,p , 0 < λ < λ

(k)
L,p has at least k pairs of sign-changing critical points

{±u(λ)
j,ε,p|1 ≤ j ≤ k}

such that

Γ(λ)
ε,p (u

(λ)
j,ε,p) = cj(ε, p, λ) ≤ c̃k, 1 ≤ j ≤ k.

Moreover, there exists x ∈ Kcj(ε,p,λ)\W with m∗(u
(λ)
j,ε,p) ≥ j, then we also have

Γλε,p(u
(λ)
j,ε,p) = Γε,p(u

(λ)
j,ε,p), DΓλε,p(u

(λ)
j,ε,p) = DΓε,p(u

(λ)
j,ε,p) = 0, j = 1, . . . , k. In this

case, we can write u
(λ)
j,ε,p as uj,ε,p and the theorem is proved. �

Assume Γε,pn(un) ≤ L,DΓε,pn(un) = 0 and pn → 2 as n → ∞. By Lemma 2.5
(2), there exists a critical point u ∈ H1(R3) of Γε, such that un → u ∈ H1(R3) up
to subsequence.

Lemma 3.7. Assume un is sign-changing critical points of Γε,pn and un → u in
H1(R3), then u is a sign-changing critical point of Γε.

Proof. Since 〈DΓε,pn(un), un〉 = 0, we have

‖un‖2H1(R3) =

∫
R3

(|∇un|2 + V (εx)u2
n) dx

≤
∫
R3

(
1

| · |
∗ upnn )upnn dx

≤ c‖un‖2pn
L

12
5 (R3)

≤ c‖un‖2pnH1(R3).

(3.5)

So there exists m > 0 such that ‖un‖H1(R3) > m and 0 6= ‖u‖H1(R3) ≥ m. Without
loss of generality, we assume that

u+ = 0 and u− 6= 0. (3.6)

We define the normalized part as

vn =
u+
n

‖u+
n ‖

. (3.7)

Then, up to a subsequence, there exists v ∈ H1(R3) such that vn ⇀ v weakly in
H1(R3). Since ∫

R3

(
1

| · |
∗ |un|pn)|vn|pn dx

≥
∫
R3

(
1

| · |
∗ |u+

n |pn)|vn|pn dx

≥
∫
R3

(|∇u+
n |2 + V (εx)(u+

n )2) dx‖u+
n ‖−pn

= ‖u+
n ‖2−pn ≥ ‖un‖2−pn → 1 as pn → 2+,

(3.8)

we have ∫
R3

(
1

| · |
∗ |u|2)|v|2 dx = lim

pn→2

∫
R3

(
1

| · |
∗ |un|pn)|vn|pn dx ≥ 1. (3.9)



12 L. YANG, X. LIU, J. ZHOU EJDE-2023/75

Then, by Hardy-Littlewood Sobolev inequality∫
R3

(
1

| · |
∗ |u|2)|v|2 dx ≤ c‖u‖2

L
12
5 (R3)

‖v‖2
L

12
5 (R3)

, (3.10)

we have ‖v‖
L

12
5 (R3)

> 0. Therefore v 6= 0 and

S = {x ∈ R3 : v(x) > 0} 6= ∅.
Since un satisfies DΓε,pn(un) = 0, in the weak sense, we have

−∆un + V (εx)un +G′
(∫

R3

χε(x)u2
n dx

)
χε(x)un

= (
1

| · |
∗ |un|pn)|un|pn−2un.

(3.11)

By elliptic regularity theory, we have un ∈ C2
loc(R3). Therefore, let x0 ∈ S, we have

un(x0) > 0 for n large enough and u(x0) = limn→∞ un(x0) ≥ 0.
On the other hand, since DΓε(u) = 0, we have

−∆u+ V (εx)u+G′
(∫

R3

χε(x)u2 dx
)
χε(x)u = (

1

| · |
∗ |u|2)u, in R3. (3.12)

Note that assumption (3.6) implies u ≤ 0 in R3. Hence, by the classical regularity
argument and the strong maximum principle on (3.12), we have u < 0 or u ≡ 0 in
R3. Since u 6= 0, we obtain u < 0 in R3. This leads to u(x0) < 0, which contradicts
u(x0) ≥ 0. Thus, the lemma is proved. �

4. Proof of Theorem 1.1

Assume un,p → un in H1(R3) as p → 2. Let εn → 0, assume DΓεn,p(un,p) =
DΓεn(un) = 0,Γεn,p(un,p),Γεn(un) ≤ L. The following two lemmas can be proved
in a similar way as in [15].

Lemma 4.1. Up to a subsequence, there exists an integer m > 0, yn,i ⊂ (Mε)
1,

yi ∈M, Ui ∈ H1(R3)\{0}, i = 1, . . . ,m such that

(1) limn→∞ |yn,i − yn,l| → ∞ if i 6= l, limn→∞ |yn,i| → ∞.

(2) yi = limn→∞ εnyn,i ∈M.
(3) For 1 ≤ i ≤ m, limn→∞ dist(yn,i, ∂Mεn) → ∞, Ui is the weak limit of

un(·+ yn,i) in H1(R3) and satisfies

−∆U + V (yi)U = (
1

| · |
∗ U2)U, U ∈ H1(R3). (4.1)

(4) limn→∞ ‖un −
∑m
i=1 Ui(· − yn,i)‖Ls(R3) = 0 for 2 < s < 2∗.

Assume that the sequence {un} satisfies the condition of Lemma 4.1, and define

Ω
(n)
R = R3\{∪mi=1B(yn,k, R)}.

Lemma 4.2. There exist c, µ independent of n, p, such that∫
Ω

(n)
R

(|∇un|2 + u2
n) dx ≤ ce−µR, ζn

∫
Ω

(n)
R

χε(x)u2
n dx ≤ ce−µR,

where ζn := G′
( ∫

R3 χε(x)u2
n dx

)
. Moreover,

|un(x)| ≤ ce−µR for x ∈ Ω
(n)
R .



EJDE-2023/75 QUADRATIC CHOQUARD EQUATIONS 13

Corollary 4.3. There exist c, µ, p1, independent of n, such that, for 2 < p < p1,
we have

|un,p(x)| ≤ c
m∑
i=1

e−µ|x−yn,i| for x ∈ R3.

Proof. The Proof of Lemma 4.2 can be done by the same method as [15, Lemma
4.3]. We only need to prove that there exists p1 > 2 such that, for 2 < p < p1,

(
1

| · |
∗ |un,p|p)up−2

n,p ≤
a

2
for x ∈ Ω

(n)
R .

By Moser’s iteration, there exists a c > 0 such that ‖un,p‖L∞(R3) ≤ c for 2 < p < p0.

By Lemma 4.2 and un,p → un in H1(R3) as p→ 2, for x ∈ Ω
(n)
R , we have

(
1

| · |
∗ |un,p|p)|un,p|p−2un,p

≤ c
∫
R3

u2
n,p

|x− y|
dyun,p

≤ c
(∫

R3

u2
n

|x− y|
dy + op(1)

)
un,p

≤ c
( 2

R

∫
|x−y|>R/2

u2
n dy + e−µR

∫
|x−y|≤R/2

1

|x− y|
dy + op(1)

)
un,p

≤ c(oR(1) + op(1))un,p .

(4.2)

Hence, there exists p1 > 2 such that for 2 < p < p1, we have cop(1) ≤ a
4 and

(
1

| · |
∗ |un,p|p)up−2

n,p ≤
a

2
for x ∈ Ω

(n)
R .

�

Lemma 4.4. If 1 ≤ i ≤ m, then y∗i = limn→∞ εnyn,i ∈ Ā.

Proof. If not, we assume that there exists i such that 1 ≤ i ≤ m and εn > 0 such
that limn→∞ εn = 0 and dist(yi, Ā) > 0. Let tk = ∇V (yi) 6= 0, by (A2) we deduce
that there exists δ1 > 0 such that

(ti,∇V (x)) ≥ 1

2
|ti|2 > 0, (ti,∇ dist(x,M)) ≥ 0 for x ∈ Bδ1(yi). (4.3)

Set

δ2 = min{|yi − yl|, yi 6= yl, i, l = 1, . . . ,m}. (4.4)

Let

0 < δ < min{1

2
δ1,

1

100
δ2}.

Denote

Bn = {x||x− yn,i| ≤ 2δε−1
n },

Tn = {x|δε−1
n ≤ |x− yn,i| ≤ 2δε−1

n }.



14 L. YANG, X. LIU, J. ZHOU EJDE-2023/75

Choose η ∈ C∞0 (R3) such that η(x) = 0 if |x−yn,i| ≥ 2δε−1
n ; η(x) = 1 if |x−yn,i| ≤

δε−1
n and |∇η| ≤ 2

δ εn(≤ 1). By 〈DΓεn(un), ϕ〉 = 0 for ϕ ∈ H1(R3), we have∫
R3

(∇un∇ϕ + V (εx)unϕ) dx+ ζn

∫
R3

χε(x)unϕdx

=

∫
R3

(
1

| · |
∗ |un|2)unϕdx,

(4.5)

where ζn = G′(
∫
R3 χε(x)u2

n dx). Choosing ϕ = tk · ∇un · η as test function in (4.5),
we obtain the local Pohoẑaev identity

1

2
εn

∫
R3

(tk,∇V (εnx))u2
nη dx+

1

2
ζn

∫
R3

(∇χεn(x), tk)u2
nη dx

=

∫
R3

(∇un,∇η)(tk,∇un) dx− 1

2

∫
R3

(|∇un|2 + u2
n)(tk,∇η) dx

− 1

2
ζn

∫
R3

χεn(x)u2
n(tk,∇η) dx+

1

2

∫
R3

(
1

| · |
∗ u2

n)u2
n(tk,∇η) dx

+
1

2

∫
R3

(∇x(
1

| · |
∗ u2

n), tk)u2
nη dx.

(4.6)

Next, we estimate all terms of (4.6). By (4.3), we have

εn

∫
R3

(tk,∇V (εnx))u2
nη dx ≥ cεn,

1

2
ζn

∫
R3

(∇χεn(x), tk)u2
nη dx ≥ 0.

Hence the left-hand side of (4.6) is greater than or equal to cεn.
Since

1

| · |
∗ u2

n =

∫
R3

u2
n(y)

|x− y|
dy

and

∇(
1

| · |
∗ u2

n) = −
∫
R3

un(y)2

|x− y|3
(x− y) dy,

we have∫
R3

(∇(
1

| · |
∗ u2

n), tk)u2
nη dx = −

∫
R3

∫
R3

u2
n(x)u2

n(y)

|x− y|3
(tk, x− y)η(x) dx dy.

Since ∫
R3

∫
R3

u2
n(x)u2

n(y)

|x− y|3
(tk, x− y)η(x)η(y) dx dy = 0 ,

we have ∫
R3

(∇(
1

| · |
∗ u2

n), tk)u2
nη dx

= −
∫
R3

∫
R3

u2
n(x)u2

n(y)

|x− y|3
(tk, x− y)η(x)(1− η(y)) dx dy.



EJDE-2023/75 QUADRATIC CHOQUARD EQUATIONS 15

Then∣∣ ∫
R3

(∇(
1

| · |
∗ u2

n), tk)u2
nη dx

∣∣ ≤ c∫∫
{|y−yn,i|≥δε

−1
n }

{|x−yn,i|≤2δε
−1
n }

u2
n(x)u2

n(y)

|x− y|2
dx dy

≤ c
∫∫
{δε−1

n ≤|y−yn,i|≤3δε
−1
n }

{|x−yn,i|≤2δε
−1
n }

u2
n(x)u2

n(y)

|x− y|2
dx dy

+ c

∫∫
{|y−yn,i|≥3δε

−1
n }

{|x−yn,i|≤2δε
−1
n }

u2
n(x)u2

n(y)

|x− y|2
dx dy

=: I + II ,

(4.7)

where

II ≤ c
∫∫
{|y−yn,i|≥3δε

−1
n }

{|x−yn,i|≤2δε
−1
n }

u2
n(y)u2

n(x) · 1

δ2
ε2
n dx dy ≤ cε2

n .

The region T̃n = {y|δε−1
n ≤ |y − yn,k| ≤ 3δε−1

n } is contained in Ω
(n)

δε−1
n

, and we have

|un(y)| ≤ ce−µδε
−1
n , y ∈ T̃n.

Then

I ≤ ce−µδε
−1
n

∫∫
{δε−1

n ≤|y−yn,k|≤3δε
−1
n }

{|x−yn,i|≤2δε
−1
n }

u2
n(x)

|x− y|2
dx dy

≤ ce−µδε
−1
n

∫∫
{|x−y|≤5δε

−1
n }

{|x−yn,i|≤2δε
−1
n }

u2
n(x)

|x− y|2
dy dx

≤ ce−µδε
−1
n ε−1

n ≤ cε2
n .

From the above estimates, by Lemma 4.2,

RHS of (4.6) ≤ c
∫
R3

(|∇un|2 + u2
n)|∇η| dx+ ζn

∫
R3

χε(x)u2
n|∇η| dx+ cε2

n

≤
∫
Tn

(|∇un|2 + u2
n)|∇η| dx+ ζn

∫
Tn

χε(x)u2
n|∇η| dx+ cε2

n

≤ ceµδε
−1
n + cε2

n ≤ cε2
n.

(4.8)

Therefore, cεn ≤ cε2
n as n → ∞. We arrived at a contradiction and completes the

proof. �

Lemma 4.5. For each δ > 0, there exists c = c(L) > 0 such that

|un(x)| ≤ ce−µ dist(x,(Aδ)εn ).

Proof. By Lemma 4.2, |un(x)| ≤ ce−µR for x ∈ Ω
(n)
R . Let Rn(x) = min{|x− yn,i| :

i = 1, . . . ,m}, then |un(x)| ≤ ce−µRn(x). Since εnyn,i → yi ∈ A, there exists ε(δ)
such that for εn ≤ ε(δ), εnyn,i ∈ Aδ, hence Rn(x) ≥ dist(x, (Aδ)εn) and

|un(x)| ≤ ce−µ dist(x,(Aδ)εn ), x ∈ R3 . (4.9)

�

Proposition 4.6. Assume DΓε(u) = 0, Γε(u) ≤ L. Then there exists ε̄ = ε̄(L)
such that Γε(u) = Iε(u) and DIε(u) = 0 if 0 < ε < ε̄.



16 L. YANG, X. LIU, J. ZHOU EJDE-2023/75

Proof. By Lemma 4.5, there exist c = c(L) and µ = µ(L) such that

|u(x)| ≤ c e−µ dist(x,(Aδ)ε) ≤ c e−µ dist(x,Mε) . (4.10)

Denote d = dist(Aδ, ∂M), then for x 6∈ Mε, we have

dist(x, (Aδ)ε) ≥ dist(x,Mε) + dε−1 . (4.11)

Hence ∫
R3

χε(x)u2 dx ≤ ε−6

∫
R3\Mε

u2 dx

≤ cε−6

∫
R3\Mε

e−2µ dist(x,(Aδ)ε) dx

≤ cε−6e−µdε
−1

∫
R3\Mε

e−µ dist(x,(Aδ)ε) dx

≤ cε−6e−µdε
−1

∫
R3\M

e−µ dist(x,Aδ) dx

≤ cε−6e−µdε
−1

→ 0 as ε→ 0 .

In particular there exists ε̄ such that for 0 < ε ≤ ε̄ we have

G
( ∫

R3

χε(x)u2 dx
)

= 0 . (4.12)

Hence, Iε(u) = Γε(u) and DIε(u) = DΓε(u) = 0. �

Lemma 4.7. There is a direct sum H1(R3) = X0 ⊕ X⊥0 such that dimX0 < ∞
and for all ϕ ∈ X⊥0 ,∫

R3

(|∇ϕ|2 + aϕ2)− c1
∫
R3

(
1

| · |
∗ e−c2|x|)ϕ2 dx− c1

∫
R3

(
1

| · |
∗ (e−c2|x|ϕ))e−c2|x|ϕdx

≥ a

2

∫
R3

ϕ2 dx ,

where c1, c2 > 0.

Proof. It suffices to prove that if X0 is a subspace of H1(R3) such that∫
R3

(|∇ϕ|2 +
a

2
ϕ2)− c1

∫
R3

(
1

| · |
∗ e−c2|x|)ϕ2 dx

− c1
∫
R3

(
1

| · |
∗ (e−c2|x|ϕ))e−c2|x|ϕdx ≤ 0, ϕ ∈ X0,

(4.13)

then X0 is finite-dimensional.∫
R3

(
1

| · |
∗ (e−c2|x|ϕ))e−c2|x|ϕdx

=

∫
R3

∫
R3

e−c2|x|ϕ(x)e−c2|y|ϕ(y)

|x− y|
dx dy

=
(∫

R3\BR(0)

∫
R3\BR(0)

+2

∫
R3\BR(0)

∫
BR(0)

+

∫
BR(0)

∫
BR(0)

)e−c2|x|ϕ(x)e−c2|y|ϕ(y)

|x− y|
dx dy

=: I1 + 2I2 + I3,

(4.14)



EJDE-2023/75 QUADRATIC CHOQUARD EQUATIONS 17

where

I1 =

∫
R3\BR(0)

∫
R3\BR(0)

e−c2|x|ϕ(x)e−c2|y|ϕ(y)

|x− y|
dx dy

≤
∫
R3\BR(0)

e−c2|x|ϕ(x)
(∫

R3\BR(0)

e−2c2|y|

|x− y|2
dy
)1/2

dx
(∫

R3\BR(0)

ϕ2 dx
)1/2

≤ c
∫
R3\BR(0)

e−c2xϕ(x) dx
(∫

R3\BR(0)

ϕ2 dx
)1/2

≤ o( 1

R
)

∫
R3\BR(0)

ϕ2 dx .

Similarly, we have

I2 ≤ o
( 1

R

) ∫
R3\BR(0)

ϕ2 dx+ c

∫
BR(0)

ϕ2 dx,

I3 ≤ c
∫
BR(0)

ϕ2 dx.

Hence, we have∫
R3

(
1

| · |
∗ (e−c2|x|ϕ))e−c2|x|ϕdx ≤ o( 1

R
)

∫
R3\BR(0)

ϕ2 dx+ c

∫
BR(0)

ϕ2 dx.

Since lim|x|→∞
1
|x| ∗ e

−c|x| = 0, we choose R > 0 such that

c

∫
R3

(
1

| · |
∗ e−c2|x|)ϕ2 dx+ c

∫
R3

(
1

| · |
∗ (e−c2|x|ϕ))e−c2|x|ϕdx

≤ a

4

∫
R3\BR(0)

ϕ2 dx+ c

∫
BR(0)

ϕ2 dx.
(4.15)

By (4.13) and (4.15), we have, for ϕ ∈ X0,∫
R3

(|∇ϕ|2 +
a

4
ϕ2) dx− c

∫
BR(0)

ϕ2 dx ≤ 0.

Hence, we obtain∫
BR(0)

|∇ϕ|2 dx+

∫
R3\BR(0)

(|∇ϕ|2 +
a

4
ϕ2) dx ≤ c

∫
BR(0)

ϕ2 dx. (4.16)

Now define the restriction operator P from L2(R3) to L2(BR(0)) by Pϕ = ϕ|BR(0).

Since (4.16) holds, it is easy to see that P is injective. Let X̃0 = PX0, it suffice to

prove X̃0 is finite-dimensional. It also follows from (4.16) that

‖ϕ‖H1(BR(0)) ≤ c‖ϕ‖L2(BR(0)), ϕ ∈ X̃0. (4.17)

Let S := {ϕ ∈ X̃0|‖ϕ‖L2(BR(0)) = 1}, then the set S is compact by (4.17). Hence,

we obtain that X̃0 is finite-dimensional subspace. �

Proposition 4.8. For each positive integer k, there exists ε′k > 0 such that for 0 <
ε < ε′k, Γε has at least k pairs of sign-changing critical points ±unj , j = 1, . . . , k.

Proof. Choose ε̃n small enough to satisfy Theorem 3.6 and Lemma 2.5. We denote
εn := {ε̃n, 1

n}. By Lemma 2.5, without loss of generality, we may assume cn(ε, p)→
cn(ε) and un,ε,p → un,ε in H1(R3) as p → 2. By Lemma 3.7, cn(ε) is a critical
value of Γε with sign-changing critical point un,ε ∈ H1(R3).



18 L. YANG, X. LIU, J. ZHOU EJDE-2023/75

We claim that for any M > 0, there is n such that cn,ε > M for any ε ∈ (0, εn).
Then for every positive integer k, we can choose (nj)

k
j=1 and ε′k = min{εnj}kj=1

such that for ε ∈ (0, ε′k), cnj ,ε > c̃j ≥ cnj−1,ε, 2 ≤ j ≤ k. As a result, we can find k

different critical values (cnj ,ε)
k
j=1 of Γε as ε ∈ (0, ε′k).

By contradiction, we assume there exists M > 0, ε̄n → ∞ as n → ∞ such that
cn := cn,ε̄n ≤ M . Hence, there exists pn ∈ (2, p0), such that cn,p := cn(ε̄n, p) ≤
M + 1 for all p ∈ (2, pn). By Theorem 3.6, let un,p ∈ H1(R3) be such that

Γε̄n,p(un,p) = cn,p ≤M + 1, DΓε̄n,p(un,p) = 0, m∗(un,p) ≥ n .
By Corollary 4.3, we have

p− 1

2

∫
R3

(
1

| · |
∗ (|un,p|p))|un,p|p−2ϕ2 dx ≤

m∑
i=1

c

∫
R3

(
1

| · |
∗ e−c|x−yn,i|)ϕ2 dx

and
p

2

∫
R3

(
1

| · |
∗ (|un,p|p−2un,pϕ))|un,p|p−2un,pϕdx

≤
m∑
i=1

c

∫
R3

( 1

| · |
∗ (e−c|x−yn,i|ϕ)

)
e−c|x−yn,i|ϕdx

+

m∑
i 6=j

c

∫
R3

( 1

| · |
∗ (e−c|x−yn,j |ϕ)

)
e−c|x−yn,i|ϕdx

≤
m∑
i=1

c

∫
R3

( 1

| · |
∗ (e−c|x−yn,i|ϕ)

)
e−c|x−yn,i|ϕdx+ o(1)

∫
R3

ϕ2 dx .

Let

Xi,n = {ϕ(x− yn,i)|ϕ ∈ X0}, Xn = {
m∑
i=1

ϕi|ϕi ∈ Xi,n}.

Then dimXi,n = dimX0, dimXn ≤ mdimX0 < ∞, H1(R3) = Xn ⊕ X⊥n , where
X⊥n = ∩mi=1X

⊥
i,n. By Lemma 4.7, for ϕ ∈ X⊥n , we have

〈D2Γε̄n,p(un,p)ϕ,ϕ〉

≥
∫
R3

|∇ϕ|2 + aϕ2 dx−
∫
R3

( 1

| · |
∗ (|un,p|p)

)
|un,p|p−2ϕ2 dx

−
∫
R3

( 1

| · |
∗ (|un,p|p−2un,pϕ)

)
|un,p|p−2un,pϕdx

≥
∫
R3

|∇ϕ|2 + aϕ2 dx−
m∑
i=1

c

∫
R3

(
1

| · |
∗ e−c|x−yn,i|)ϕ2 dx

−
m∑
i=1

c

∫
R3

(
1

| · |
∗ (e−c|x−yn,i|ϕ))e−c|x−yn,i|ϕdx− o(1)

∫
R3

ϕ2 dx

≥ (
a

2
− o(1))

∫
R3

ϕ2 dx

≥ a

4

∫
R3

ϕ2 dx .

As a result, we can get m∗(un,p) ≤ m dimX0 ≤ C, for some C > 0 independent of
n, which contradicts to m∗(un,p) ≥ n→∞. �



EJDE-2023/75 QUADRATIC CHOQUARD EQUATIONS 19

The proof Theorem 1.1 follows from Propositions 4.6 and 4.8.

Acknowledgments. The authors are grateful to the referees for their helpful com-
ments. This work is supported by NSFC 12161093 and 11961078, and by the Yun-
nan Key Laboratory of Modern Analytical Mathematics and Applications, Yunnan
Province, China.

References

[1] N. Ackermann; On a periodic Schrödinger equation with nonlocal superlinear part. Math. Z.,

248 (2004), 423–443.
[2] C. O. Alves, H. Luo, M. Yang; Ground state solutions for a class of strongly indefinite

Choquard equations. Bull. Malays. Math. Sci. Soc., 43 (2020), 3271–3304.

[3] C. O. Alves, A. B. Nóbrega, M. Yang; Multi-bump solutions for Choquard equation with
deepening potential well. Calc. Var. Partial Differential Equations, 55 (2016), 1–28.

[4] J. Byeon, Z. Q. Wang; Standing waves with a critical frequency for nonlinear Schrödinger
equations. Arch. Ration. Mech. Anal., 165 (2002), 295–316.

[5] D. Cassani, J. Van Schaftingen, J. Zhang; Ground states for Choquard type equations with

Hardy-Littlewood-Sobolev lower critical exponent. Proc. Roy. Soc. Edinburgh Sect. A, 150
(2020), 1377–1400.

[6] S. Chen, Z. Q. Wang; Localized nodal solutions of higher topological type for semiclassical

nonlinear Schrödinger equations. Calc. Var. Partial Differential Equations, 56 (2017), 1–26.
[7] S. Cingolani, K. Tanaka; Semi-classical states for the nonlinear Choquard equations: ex-

istence, multiplicity and concentration at a potential well. Rev. Mat. Iberoam., 35 (2019),

1885–1924.
[8] M. Clapp, D. Salazar; Positive and sign changing solutions to a nonlinear Choquard equation.

J. Math. Anal. Appl., 407 (2013), 1–15.

[9] F. Gao, E. D. da Silva, M. Yang, J. Zhou; Existence of solutions for critical Choquard
equations via the concentration-compactness method. Proc. Roy. Soc. Edinburgh Sect. A,

150 (2020), 921–954.
[10] F. Gao, M. Yang; On nonlocal Choquard equations with Hardy-Littlewood-Sobolev critical

exponents. J. Math. Anal. Appl., 448 (2017), 1006–1041.

[11] M. Ghimenti, V. Moroz, J. Van Schaftingen; Least action nodal solutions for the quadratic
Choquard equation. Proc. Amer. Math. Soc., 145 (2017), 737–747.

[12] M. Ghimenti, J. Van Schaftingen; Nodal solutions for the Choquard equation. J. Funct.

Anal., 271 (2016), 107–135.
[13] C. Gui, H. Guo; On nodal solutions of the nonlinear Choquard equation. Adv. Nonlinear

Stud., 19 (4)(2019), 677–691.

[14] L. Guo, T. Hu, S. Peng, W. Shuai; Existence and uniqueness of solutions for Choquard equa-
tion involving Hardy-Littlewood-Sobolev critical exponent. Calc. Var. Partial Differential
Equations, 58 (2019), 1–34.

[15] R. He, X. Liu; Localized nodal solutions for semiclassical Choquard equations. J. Math.
Phys., 62 (2021), 091511.

[16] Z. Huang, J. Yang, W. Yu; Multiple nodal solutions of nonlinear Choquard equations. Elec-
tron. J. Differential Equations, 268 (2017), 1–18.

[17] X. Li, S. Ma; Choquard equations with critical nonlinearities. Commun. Contemp. Math., 22
(2020), 1950023, 28.

[18] E. H. Lieb; Existence and uniqueness of the minimizing solution of Choquard’s nonlinear
equation. Studies Appl. Math., 57 (1976/77), 93–105.

[19] E. H. Lieb, M. Loss; Analysis, second ed., in Graduate Studies in Mathematics, vol. 14,
American Mathematical Society, Providence, RI, 2001.

[20] P. L. Lions; The Choquard equation and related questions. Nonlinear Anal., 4(1980), 1063–
1072.

[21] J. Liu, X. Liu, Z. Q. Wang; Sign-changing solutions for coupled nonlinear Schrödinger equa-
tions with critical growth. J. Differential Equations, 261 (2016), 7194–7236.

[22] J. Liu, X. Liu, Z. Q. Wang; Multiple mixed states of nodal solutions for nonlinear Schrödinger
systems. Calc. Var. Partial Differential Equations, 52 (2015), 565–586.



20 L. YANG, X. LIU, J. ZHOU EJDE-2023/75

[23] X. Liu; Localized nodal solutions for system of critical Choquard equations. Commun. Non-

linear Sci. Numer. Simul., 121 (2023), 107190.

[24] X. Liu, J. Liu, Z. Q. Wang; Quasilinear elliptic equations via perturbation method. Proc.
Amer. Math. Soc., 141 (2013), 253–263.

[25] L. Ma, L. Zhao; Classification of positive solitary solutions of the nonlinear Choquard equa-

tion. Arch. Ration. Mech. Anal., 195 (2010), 455–467.
[26] V. Moroz, J. Van Schaftingen; Groundstates of nonlinear Choquard equations: existence,

qualitative properties and decay asymptotics. J. Funct. Anal., 265 (2013), 153–184.

[27] V. Moroz, J. Van Schaftingen; Groundstates of nonlinear Choquard equations: Hardy-
Littlewood-Sobolev critical exponent. Commun. Contemp. Math., 17 (2015), 1550005, 12.

[28] V. Moroz, J. Van Schaftingen; Semi-classical states for the Choquard equation. Calc. Var.

Partial Differential Equations, 52 (2015), 199–235.
[29] I. M. Moroz, R. Penrose, P. Tod; Spherically-symmetric solutions of the Schrödinger-Newton

equations. Class. Quantum Gravity, 15 (1998), 2733–2742.
[30] S. I. Pekar; Untersuchungen über die elektronentheorie der kristalle. De Gruyter, 1954.

[31] D. Qin, V. D. Rădulescu, X. Tang; Ground states and geometrically distinct solutions for

periodic Choquard-Pekar equations. J. Differential Equations, 275 (2021), 652–683.
[32] D. Ruiz, J. Van Schaftingen; Odd symmetry of least energy nodal solutions for the Choquard

equation. J. Differential Equations, 264 (2018), 1231–1262.

[33] K. Tintarev, K. H. Fieseler; Concentration compactness. Functional-analytic grounds and
applications. Imperial College Press, London, 2007.

[34] B. Zhang, X. Liu; Localized nodal solutions for semiclassical quasilinear Choquard equations

with subcritical growth. Electron. J. Differential Equations, 11 (2022), 1–29.
[35] C. L. Xiang; Uniqueness and nondegeneracy of ground states for Choquard equations in three

dimensions. Calc. Var. Partial Differential Equations, 55 (2016), 1–25.

[36] C. X. Zhang, Z.-Q. Wang. Concentration of nodal solutions for logarithmic scalar field equa-
tions. J. Math. Pures Appl., 135 (2020), 1–25.

Lu Yang

Department of Mathematics, Yunnan University, Kunming, 650091, China
Email address: 2973434415@qq.com

Xiangqing Liu
Department of Mathematics, Yunnan Normal University, Kunming, 650500, China

Email address: lxq8u8@163.com

Jianwen Zhou (corresponding author)
Department of Mathematics, Yunnan University, Kunming, 650091, China

Email address: jwzhou@ynu.edu.cn


	1. Introduction
	2. (PS)c condition for ,p
	3. Existence of sign-changing critical points for 
	4. Proof of Theorem 1.1
	Acknowledgments

	References