Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 81, pp. 1–17.

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

MIXED LOCAL AND NONLOCAL SCHRÖDINGER-POISSON

TYPE SYSTEM INVOLVING VARIABLE EXPONENTS

XIAOLU LIN, SHENZHOU ZHENG

Abstract. We consider the existence of solutions for a class of Schrödinger-

Poisson type equations with mixed local and nonlocal p-Laplacian. More pre-

cisely, we obtain two distinct nontrivial solutions for the problem involving
variable exponents growth by the variational methods. Moreover, the phe-

nomena of concentration and multiplicity of solutions are also investigated as

λ→ ∞.

1. Introduction

Let Ω ⊂ RN be a bounded domain with smooth boundary and let α, β be two
positive parameters. The purpose of this article is to investigate the existence
and asymptotic behavior, as λ → ∞, of solutions of the Schrödinger-Poisson type
system

Lu+ λV (x)|u|p−2u+ φ|u|p−2u = α|u|p(x)−2u+ β|u|q(x)−2u in Ω,

−∆φ = up in Ω,

u = φ = 0 in RN \ Ω,

(1.1)

where λ is a positive parameter, and V (x) ∈ C(RN ) is a potential function. Here,
α|u|p(x)−2u+β|u|q(x)−2u is assumed to be a concave-convex nonlinearity with vari-
able functions p(x), q(x) ∈ C(Ω). We stress that the operator L appearing (1.1)
represents the superposition of a p-Laplacian and a fractional p-Laplacian, defined
as

L := −∆p + (−∆)sp for some s ∈ (0, 1),

where ∆pu = div(|∇u|p−2∇u) and (−∆)sp is the nonlocal operator given by

(−∆)spϕ = P.V.

∫
RN

|ϕ(x)− ϕ(y)|p−2
(
ϕ(x)− ϕ(y)

)
|x− y|N+ps

dy.

Here P.V. denotes the principle value of the integral. See [15, 25] for further details
on the fractional p-Laplacian.

The operator L models diffusion patterns over a variety of time scales. So it
is used in applications such as of optimal search methods, biomathematics, and
animal foraging; see for example [16, 17] and their references. The mixed local

2020 Mathematics Subject Classification. 35A15, 35B33, 35R11.
Key words and phrases. Mixed local and nonlocal operator; Schrödinger-Poisson type system;

multiple solutions; asymptotic behavior; symmetric mountain pass lemma.
©2022. This work is licensed under a CC BY 4.0 license.

Submitted May 4, 2022. Published November 28, 2022.

1



2 X. LIN, S. ZHENG EJDE-2022/81

and nonlocal operator L is remarkably similar to the mixed (s, t)-order operator in
mathematical research. For the significance, applications and some results of mixed
(s, t)-order operators, see see [27, 33].

Problem (1.1) will play a crucial part while associated with standing wave solu-
tions ψ(x, t) = u(x)e−ıt to the time-dependent Schrödinger-Poisson system

−i∂ψ
∂t

= −∆ψ + φ(x)ψ − f(ψ) in Ω,

−∆φ = |ψ|2 in Ω,

ψ = φ = 0 on ∂Ω.

(1.2)

This system is employed in quantum mechanics to describe the interaction of a
charged particle with an electrostatic field, where ψ and φ represent the wave
functions connected to the particle and the electric potentials, respectively. The
nonlinearity f(ψ) is usually used to model the interaction between multiple parti-
cles. For more information on the physical background we refer the readers to [32].
Under various assumptions, the existence of solution for (1.2) have received a lot
of interest in recent years.

Existence, nonexistence, and asymptotic behavior of solutions to Schrödinger-
Poisson systems have been studied by Du, Tian, Wang and Zhang [18]. This is
done under suitable assumptions on potential well V (x) and the nonlinearity f(·)
(and asymptotically 3-linear), using variational methods. Recently, Jeanjean and
Le [23] obtained multiple normalized solutions for Sobolev critical Schrödinger-
Poisson-Slater equation. For more results see [1, 21].

Let us now review topics associated with the so-called concave-convex nonlin-
earity. The emergence of substantial literature on concave-convex nonlinear elliptic
problems started with the pioneering work by Ambrosetti, Brezis and Cerami [4],
for Laplacian problem in a bounded domain,

−∆z = λ|z|q−2z + |z|h−2z in Ω,

z = 0 on ∂Ω,
(1.3)

where 1 < q < 2 < h < 2∗. Brändle et al. [9] studied the concave-convex-type ellip-
tic problems driven by a nonlocal integro-differential operator. Subsequently, Ho
and Sim [22] discussed the multiplicity of nontrivial solutions to degenerate p(x)-
Laplacian equations involving concave-convex type nonlinearities with two parame-
ters. For studying variable exponential problems, there are two main reasons: One
is that they frequently occur in many different applications, including electrorhe-
ological fluids, image processing and elastic mechanics, see [12, 29]. The other
reason is from a pure mathematical standpoint, as variable exponents have more
complex nonlinearities than problems with constant exponents. For more results on
variable exponent problems see [2, 7, 11]. Regarding the unbalanced double-phase
problem with variable exponent, we would like to remark that Kim et al. [24] ob-
tained L∞-bound of solutions via the De Giorgi iteration method and a truncated
energy technique. Related to double-phase problem, we refer to [14, 26, 37] and
their references.

For the Schrödinger-Poisson system with concave and convex terms

−∆u+ V (x)u+ µφu = a(x)|u|p−2u+ b(x)|u|q−2u in R3,

−∆φ = u2 in R3,
(1.4)



EJDE-2022/81 SCHRÖDINGER-POISSON SYSTEM WITH VARIABLE EXPONENTS 3

Sun, Su, and Zhao [30] showed that it has infinitely many solutions, under the
constraint that φ(x)→ 0 as |x| → +∞. By developing a novel constraint approach,
Sun and Wu [31] showed that (1.4) has at least two positive solutions, under the
assumption that V (x) satisfying a steep potential well condition. By quantitative
deformation lemma, Yang and Ou [36] proved that the Schrödinger-Poisson system
with concave-convex nonlinearity admits a nodal solution in a bounded domain.

Inspired by the above facts, our main goal is to consider the Schrödinger-Poisson
system for mixed order p-Laplacian with variable exponent growth. More precisely,
we show that there exist two distinct solutions of (1.1), and then discuss the con-
centration and multiplicity of solutions as λ → ∞. To the best of our knowledge,
this is the first attempt to investigate the existence of solution for mixed order
Schrödinger-Poisson system involving variable exponent growth.

To obtain our main results, we use the following assumptions: On the continuous
potential function V : Ω→ [0,∞), we assume that

(A1) Z = int(V −1(0)) ⊂ Ω is a nonempty bounded domain;
(A2) there exists a nonempty open domain Ω0 ⊂ Z such that V (x) ≡ 0 for all

x ∈ Ω0.

On the variable exponents p(x), q(x) ∈ C(Ω) we assume that

(A3) 2p < p(x) < Np
N−ps for all x ∈ Ω;

(A4) 1 < q(x) < p for all x ∈ Ω;
(A5) Let α and β be two positive parameters such that

α ≤ p− q+

Ap(p+ − q+)
, αp−q

+

βp
+−p ≤

( p− q+

Ap(p+ − q+)

)p−q+( p+ − p
Bp(p+ − q+)

)p+−p
,

where constants A,B will be specified in Lemma 3.3.

Our first main results reads as follows.

Theorem 1.1. Assume (A1)–(A5) hold. Then for all λ > 0, Problem (1.1) has at
least two distinct nontrivial solutions.

To be precise, Theorem 1.1 shows that (1.1) has a positive energy solution u1
λ

and a negative energy solution u2
λ. The basic strategies for proving Theorem 1.1

were used by Alves and Ferreira [2], and are based on the mountain pass theorem
(cf. [5]) and Ekeland variational principle (cf.[20]). The process in [2] does not
seem to be entirely applicable in our setting because our consideration involves a
nonlocal term φ and a mixed order operator.

The following result is associated with the concentration behavior of the solutions
from Theorem 1.1 as λ→∞.

Theorem 1.2. Let u1
λ and u2

λ be two solutions of (1.1) from Theorem 1.1. Suppose

that (A1)—(A5) hold, then there exists u1, u2 ∈ W 1,p
0 (Ω) such that u1

λ → u1 and

u2
λ → u2 in W 1,p

0 (Ω) as λ→∞. Moreover, u1 6= u2 are two nontrivial solutions of
the problem

Lu+ φ(x)|u|p−2u = α|u|p(x)−2u+ β|u|q(x)−2u in Ω0,

−∆φ = |u|p in Ω0,

u = φ = 0 in RN \ Ω0.

(1.5)



4 X. LIN, S. ZHENG EJDE-2022/81

Theorem 1.2 is proved by analyzing the convergence property of u1
λ and u2

λ as
λ→∞. In this process, we need to confirm that u1 = u2 = 0 in RN \Ω0, which is
fulfilled by the idea of Bartsh, Pankov and Wang in [8].

Finally, we are to establish the existence of infinitely many solutions for (1.5).

Theorem 1.3. Assume that (A3)–(A5) hold. Then (1.5) admits infinitely many
solutions.

The rest of this article is organized as follows. In Section 2, the variational
framework and some preliminaries are recalled. We devote Section 3 to two distinct
nontrivial weak solutions for the problem (1.1) by using mountain pass theorem and
Ekeland variational principle. In Section 4, the concentration of the weak solutions
of the problem (1.1) is considered. Finally, we focus on the existence of infinitely
many solutions of the problem (1.5) based on the symmetric mountain pass theorem
in Section 5.

2. Preliminaries

In this section we introduce a functional framework related to problem (1.1).
Let us begin with reviewing some notation and valuable conclusion concerning the
variable exponent Lebesgue spaces that will be used later. Throughout this article,
we assume that p(x) ∈ C(Ω) : Ω→ (1,∞) and denote

p− := ess infx∈Ω p(x), p+ := ess supx∈Ω p(x).

The variable exponent Lebesgue space is defined by

Lp(x)(Ω) =
{
u : Ω→ R is a measurable function; ρp(x)(u) =

∫
Ω

|u(x)|p(x)dx <∞
}

with the Luxemburg norm

‖u‖Lp(x)(Ω) = inf
{
µ > 0 : ρp(x)(µ

−1u) =

∫
Ω

∣∣u(x)

µ

∣∣p(x) ≤ 1
}
.

Here, p(x) is said to be bounded if p+ is finite. For this case, it is easy to see that

‖u‖p
−

Lp(x)(Ω)
≤ ρp(x)(u) ≤ ‖u‖p

+

Lp(x)(Ω)
if ‖u‖Lp(x)(Ω) ≥ 1,

‖u‖p
+

Lp(x)(Ω)
≤ ρp(x)(u) ≤ ‖u‖p

−

Lp(x)(Ω)
if ‖u‖Lp(x)(Ω) ≤ 1.

Definition 2.1. The dual space of Lp(x)(Ω) is Lp
′(x)(Ω), where the conjugate

exponent p(x)′ is defined as p(x)′ = p(x)/(p(x)− 1).

If 1 < p− ≤ p+ < ∞, then the variable exponent Lebesgue space Lp(x)(Ω) is
a reflexive uniformly convex Banach space. Obviously, for Lp(x)(Ω), the Hölder
inequality is still valid.

Lemma 2.2 ([28]). For all u ∈ Lp(x)(Ω), v ∈ Lp′(x)(Ω) with p(x) ∈ (1,∞), it holds∫
Ω

|uv|dx ≤
( 1

p−
+

1

(p′)−

)
‖u‖Lp(x)(Ω)‖v‖Lp′(x)(Ω) ≤ 2‖u‖Lp(x)(Ω)‖v‖Lp′(x)(Ω).

For more information on variable exponent Lebesgue spaces, we refer the readers
to [28]. For any 1 < p < ∞, W 1,p(Ω) is the usual Sobolev space equipped with
norm

‖u‖W 1,p(Ω) := ‖u‖Lp(Ω) + ‖∇u‖Lp(Ω).



EJDE-2022/81 SCHRÖDINGER-POISSON SYSTEM WITH VARIABLE EXPONENTS 5

The closure of C∞c (Ω) inW 1,p(Ω) is denoted byW 1,p
0 (Ω). In what follows, we review

several fundamental results related to the fractional Sobolev space W s,p(RN ), for
more details see [15]. Before we do that, let us recall the so-called Gagliardo
seminorm of u

[u]s,p =
(∫∫

R2N

|u(x)− u(y)|p

|x− y|N+ps
dx dy

)1/p

.

Definition 2.3. Let u be a measurable function, and let

W s,p(RN ) :=
{
u ∈ Lp(RN ) :

∫∫
R2N

|u(x)− u(y)|p

|x− y|N+ps
dx dy <∞

}
endowed with the norm

‖u‖W s,p(RN ) =
(∫

RN
|u(x)|pdx+

∫∫
R2N

|u(x)− u(y)|p

|x− y|N+ps
dx dy

)1/p

.

The space W s,p(Ω) is defined similarly by confining to a domain Ω. Obviously,
the fractional Sobolev space with zero boundary value

W s,p
0 (Ω) =

{
u ∈W s,p(RN ) : u = 0 on RN \ Ω

}
is a reflexive Banach space. The next result asserts that W 1,p(Ω) is continuously
embedded in W s,p(Ω), see [15, Proposition 2.2].

Lemma 2.4. For 0 < s < 1 < p < ∞, there exists a constant C = C(N, p, s) > 0
such that

‖u‖W s,p(Ω) ≤ C‖u‖W 1,p(Ω), ∀u ∈W 1,p(Ω).

The simultaneous existence of the local operator ∆p enables us to work in a
simpler space. More precisely, see [10, Lemma 2.1].

Lemma 2.5. There exists a constant C = C(N, p, s,Ω) such that

[ū]ps,p :=

∫∫
R2N

|ū(x)− ū(y)|p

|x− y|N+ps
dx dy ≤ C

∫
Ω

|∇u|pdx ∀u ∈W 1,p
0 (Ω),

where ū is the extension as zero of u in all RN .

For a zero extension with u = 0 in RN \Ω in (1.1), it does not matter if we write

[ū]s,p as [u]s,p. Moreover, the mixed norm on the space W 1,p
0 (Ω) is denoted as

‖u‖W 1,p
0 (Ω) :=

(∫
Ω

|∇u|pdx+

∫∫
R2N

|u(x)− u(y)|p

|x− y|N+ps
dx dy

)1/p

.

For a proof of the following Sobolev embedding we refer the reader to [15].

Lemma 2.6. The embedding

W 1,p
0 (Ω) ↪→


Lt(Ω) for t ∈ [1, p∗], if p ∈ (1, N) ,

Lt(Ω) for t ∈ [1,∞], if p = N,

L∞(Ω) if p > N,

are continuous. More precisely, the above embeddings are compact, except for t =
p∗ = Np

N−p , if p ∈ (1, N).



6 X. LIN, S. ZHENG EJDE-2022/81

In regards to the existence of the potential V (x) in Problem (1.1), it is necessary
to introduce the function spaces

E =
{
u ∈W 1,p

0 (Ω) : [u]ps,p +

∫
Ω

|∇u|pdx+

∫
Ω

V (x)|u|pdx <∞
}

equipped with the inner product

〈u, v〉E =

∫∫
R2N

|u(x)− u(y)|p−2(u(x)− u(y))(v(x)− v(y))

|x− y|N+ps
dx dy

+

∫
Ω

|∇u|p−2uv dx+

∫
Ω

V (x)|u|p−2uv dx for all u, v ∈ E

and the corresponding norm ‖u‖pE = 〈u, u〉E . For λ > 0, we also need the inner
product

〈u, v〉λ =

∫∫
R2N

|u(x)− u(y)|p−2(u(x)− u(y))(v(x)− v(y))

|x− y|N+ps
dx dy

+

∫
Ω

|∇u|p−2uv dx+ λ

∫
Ω

V (x)|u|p−2uv dx for all u, v ∈ E

and the corresponding norm ‖u‖pλ = 〈u, u〉λ. Clearly, ‖u‖E ≤ ‖u‖λ for all λ ≥ 1.
Obviously, Eλ = (E, ‖ · ‖λ) is a reflexive Banach space.

Lemma 2.7. Let p(x) : Ω→ R be a continuous function satisfying (A3) and p(x) ∈
(1, Np/(N − ps)) for any x ∈ Ω. Then, the embedding Eλ ↪→ Lp(x)(Ω) is continuous
and compact. More precisely, there exist a constant Cp = C(N, p+, p, s) > 0 such
that ∫

Ω

|u(x)|p(x)dx ≤ max
{
‖u‖p

+

Lp(x)
, ‖u‖p

−

Lp(x)

}
≤ max

{
Cp

+

p [u]p
+

s,p, C
p−

p [u]p
−

s,p

}
≤ max

{
Cp

+

p ‖u‖
p+

λ , Cp
−

p ‖u‖
p−

λ

}
.

(2.1)

It is important that System (1.1) can be reduced into one single Schrödinger

equation with a nonlocal term, see for instance [3, 6]. For every fixed u ∈W 1,p
0 (Ω),

by applying the so-called Newton potential, we find a function φu ∈ H1
0 (Ω) satis-

fying

−∆φ = |u|p in Ω,

φ = 0 in RN \ Ω.

Then, we use a standard argument to obtain that φu satisfies the following proper-
ties, see [19].

Lemma 2.8. For any u ∈W 1,p
0 (Ω), we have

(1) φu ≥ 0 and φtu = tpφu for any t ≥ 0;
(2) there exists a constant C > 0 such that

‖∇φu‖L2(Ω) =

∫
Ω

φu|u|pdx ≤ C‖u‖pλ;

(3) if un ⇀ u in W 1,p
0 (Ω), then φun ⇀ φu in H1

0 (Ω) and

lim
n→∞

∫
Ω

φunu
p−2
n unφdx =

∫
Ω

φuu
p−2uφ dx for all φ ∈W 1,p

0 (Ω).



EJDE-2022/81 SCHRÖDINGER-POISSON SYSTEM WITH VARIABLE EXPONENTS 7

Putting φ = φu into the first equation of (1.1), our problem (1.1) reduces to the
single fractional Schrödinger equation

Lu+ λV (x)|u|p−2u+ φu|u|p−2u = α|u|p(x)−2u+ β|u|q(x)−2u in Ω,

u = 0 in RN \ Ω.
(2.2)

We are now in a position to give the definition of weak solution to (2.2).

Definition 2.9. We say that u ∈ Eλ is a weak solution of (2.2), if u satisfies∫∫
R2N

|u(x)− u(y)|p−2(u(x)− u(y))(v(x)− v(y))

|x− y|N+ps
dx dy

+

∫
Ω

|∇u|p−2uv dx+

∫
Ω

λV (x)|u|p−2uv + φu|u|p−2uv dx

=

∫
Ω

(
α|u|p(x)−2uv + β|u|q(x)−2uv

)
dx

for any v ∈ Eλ.

Note that, if u is the solution of Schrödinger equation (2.2), then u is the solution
of (1.1). We would like to emphasize that the existence of solutions for (2.2) can
be established by a variational method. Clearly, the energy functional Iλ : Eλ → R
associated with Problem (2.2) is

Iλ(u) =
1

p

∫
Ω

|∇u|pdx+
1

p

∫∫
R2N

|u(x)− u(y)|p

|x− y|N+ps
dx dy +

λ

p

∫
Ω

V (x)|u|pdx

+
1

2p

∫
Ω

φu|u|pdx−
∫

Ω

( α

p(x)
|u|p(x) +

β

q(x)
|u|q(x)

)
dx

=
1

p
‖u‖pλ +

1

2p

∫
Ω

φu|u|pdx−
∫

Ω

( α

p(x)
|u|p(x) +

β

q(x)
|u|q(x)

)
dx.

Then, we employ the argument used in [34] to prove that Iλ(u) is well-defined, of
class C1 in Eλ and

〈I ′λ(u), v〉 = 〈u, v〉λ +

∫
Ω

φu|u|p−2uv dx−
∫

Ω

( α

p(x)
|u|p(x) +

β

q(x)
|u|q(x)

)
dx

for all u, v ∈ Eλ. Hence, if u ∈ Eλ is a critical point of the functional Iλ, then it
leads to u being a solution of (2.2).

3. Proof of Theorem 1.1

To show the existence of solutions for (2.2), let us first recall the definition of
Palais-Smale sequence.

Definition 3.1. A sequence {un}n ⊂ Eλ is called a (PS)c sequence, if J(un)→ c
and J ′(un)→ 0. We say J satisfies (PS)c condition if any (PS)c sequence admits
a converging subsequence.

The following general mountain pass theorem (cf. [5]), which allows us to find a
(PS)c sequence.

Theorem 3.2. Let X be a real Banach space, and J ∈ C1(X,R) with J(0) = 0.
Suppose that

(i) there exist two constants ρ, δ > 0 such that J(u) ≥ δ for u ∈ X with
‖u‖X = ρ;



8 X. LIN, S. ZHENG EJDE-2022/81

(ii) there exists an e ∈ X satisfying ‖e‖X > ρ such that J(e) < 0.

If we define Γ =
{
γ ∈ C1([0, 1];X) : γ(0) = 0, γ(1) = e

}
. Then

c = inf
γ∈Γ

max
0≤t≤1

J
(
γ(t)

)
≥ δ,

and there exists a (PS)c sequence {un}n ⊂ X.

Before using the mountain pass theorem to prove Theorem 1.1, we verify that
Iλ possesses the mountain pass geometry (i) and (ii).

Lemma 3.3. Assume that (A1)–(A5) hold. Then for all λ > 0, there exists two
positive constants δ and ρ such that Iλ(u) ≥ δ > 0, for any u ∈ Eλ with ‖u‖λ = ρ,
where δ is independent of λ.

Proof. Using (A3)–(A4) and (2.1), for all u ∈ Eλ, we conclude that∫
Ω

( α

p(x)
|u|p(x) +

β

q(x)
|u|q(x)

)
dx

≤ α

p−

∫
Ω

|u|p(x)dx+
β

q−

∫
Ω

|u|q(x)dx

≤ α

p−
max

{
Cp

+

p ‖u‖
p+

λ , Cp
−

p ‖u‖
p−

λ

}
+

β

q−
max

{
Cq

+

q ‖u‖
q+

λ , Cq
−

q ‖u‖
q−

λ

}
,

(3.1)

where Cq, Cp > 0 are the constants defined in Lemma 2.7. Now, we deduce from
Lemma 2.8-(1) and (3.1) that

Iλ(u) ≥ 1

p
‖u‖pλ −

α

p−
max

{
Cp

+

p , Cp
−

p

}
‖u‖p

+

λ −
β

q−
max

{
Cq

+

q , Cq
−

q

}
‖u‖q

+

λ

for all u ∈ Eλ with ‖u‖λ ≥ 1.
Next, let us introduce two functions Φ(t) : [0,∞)→ R and Ψ(t) : [0,∞)→ R as

follows:

Φ(t) = Ψ(t)tq
+

, Ψ(t) =
1

p
tp−q

+

−Aαtp
+−q+ −Bβ;

where

A :=
max{Cp+p , Cp

−

p }
p−

> 0, B :=
max{Cq+q , Cq

−

q }
q−

> 0.

According to assumption (A5),

Bβ <
( p− q+

Ap(p+ − q+)

) p−q+
p+−p p+ − p

p(p+ − q+)
.

It is easy to check that Ψ(t) attains its maximum at

t = t∗ = [
p− q+

Aαp(p+ − q+)
]

1

p+−p ;

that is, Ψ(t∗) = maxt≥0 Ψ(t) > 0. Meanwhile, one easily ensures that

t∗ =
[ p− q+

Aαp(p+ − q+)

] 1

p+−p ≥ 1.

provided that

α ≤ p− q+

Ap(p+ − q+)
,

which is guaranteed by Condition (A5). Hence, the conclusion follows by letting
ρ = t∗ > 0 and δ = Φ(t∗) > 0. �



EJDE-2022/81 SCHRÖDINGER-POISSON SYSTEM WITH VARIABLE EXPONENTS 9

Lemma 3.4. Suppose that (A1)–(A5) hold. Then there exists an e ∈ Eλ with
‖e‖λ > ρ such that Iλ(e) < 0 for all λ > 0, where ρ > 0 is obtained in Lemma 3.3.

Proof. Choosing a function u0 ∈ Eλ such that

‖u0‖λ = 1 and

∫
Ω

|u0|p(x)dx > 0.

Then using Lemma 2.8-(2), we have

Iλ(tu0) =
tp

p
‖u0‖pλ +

t2p

2p
‖∇φu0

‖2L2(Ω) −
∫

Ω

( α

p(x)
|tu0|p(x) +

β

q(x)
|tu0|q(x)

)
dx

≤ tp

p
‖u0‖pλ +

t2p

2p
‖u0‖2pλ −

α

p(x)

∫
Ω

|tu0|p(x)dx

≤ tp

p
+
t2p

2p
− αtp

−

p+

∫
Ω

|u0|p(x)dx.

Thus, considering p < 2p < p− we see that there exists t0 ≥ 1 large enough such
that ‖t0u0‖λ > ρ and Iλ(t0u0) < 0. The proof is completed by letting e = t0u0. �

With Lemma 3.3, Lemma 3.4, and Theorem 3.2 in hand, for all λ > 0, the
(PS)cλ sequence of the functional Iλ(u) at the level

cλ := inf
γ∈Γ

max
0≤t≤1

Iλ
(
γ(t)

)
≥ δ > 0 (3.2)

can be constructed, where the set of paths is defined by

Γ = {γ ∈ C1([0, 1];Eλ) : γ(0) = 0, γ(1) = e}.

Lemma 3.5. Assume that (A1)–(A5) hold. If {un}n ⊂ Eλ is a (PS) sequence,
then there exists C > 0 such that ‖un‖λ ≤ C for all λ > 0.

Proof. Let {un}n∈N ⊂ Eλ be a Palais-Smale sequence of the functional Iλ, which
implies that

c+ o(1) =
1

p
‖un‖pλ +

1

2p

∫
Ω

φun |un|pdx−
∫

Ω

α

p(x)
|un|p(x) +

β

q(x)
|un|q(x)dx (3.3)

and

o(1) = 〈I ′λ(un), un〉 = ‖un‖pλ +

∫
Ω

φun |un|pdx−
∫

Ω

α|un|p(x) + β|un|q(x)dx. (3.4)

Taking into account (A3), (A4), and (2.1), we obtain that

c+ o(1)

= Iλ(un)− 1

p−
〈I ′λ(un), un〉

=
(1

p
− 1

p−

)
‖un‖pλ +

( 1

2p
− 1

p−

)∫
Ω

φun |un|pdx

−
∫

Ω

α
( 1

p(x)
− 1

p−

)
|un|p(x) −

∫
Ω

β
( 1

q(x)
− 1

p−

)
|un|q(x)dx

≥
(1

p
− 1

p−

)
‖un‖pλ − β

( 1

q−
− 1

p−

)∫
Ω

|un|q(x)dx

≥
(1

p
− 1

p−

)
‖un‖pλ − β

( 1

q−
− 1

p−

)
max

{
Cq

+

q ‖un‖
q+

λ , Cq
−

q ‖un‖
q−

λ

}
.

(3.5)



10 X. LIN, S. ZHENG EJDE-2022/81

Next we argue by contradiction. We assume that {un}n is not bounded in Eλ.
Then there exists a subsequence still denoted by {un}n such that ‖un‖λ → ∞ as
n→∞. Thus, from (3.5) it holds that

c+ o(1)

‖un‖pλ
≥
(1

p
− 1

p−

)
− β

( 1

q−
− 1

p−

)
max

{
Cq

+

q ‖un‖
q+−p
λ , Cq

−

q ‖un‖
q−−p
λ

}
,

which contradicts that q− < q+ < p < p− < p+. This completes the proof. �

Lemma 3.6. Suppose that (A1)–(A5) hold. Then the functional Iλ satisfies the
(PS)c condition in Eλ for all c ∈ R and λ > 0.

Proof. Let us choose a Palais-Smale sequence {un}n ⊂ Eλ of Iλ with c ∈ R, which
up to a subsequence, is bounded via Lemma 3.5. Thus, there exists a function
u ∈ Eλ such that

un ⇀ u in Eλ,

un → u a.e. in R,

|un|p(x)−2un ⇀ |u|p(x)−2u in Lp
′(x)(Ω).

(3.6)

Next we prove that un → u in Eλ. In fact, as the first matter of all, we use the
Hölder inequality and Lemma 2.8(3) to infer that∫

Ω

(φun |un|p−2un − φu|u|p−2u)(un − u)dx

≤ ‖φun‖L2∗ (Ω)

∥∥ |un|p−2un(un − u)
∥∥
L(2∗−1)/2∗ (Ω)

+ ‖φu‖L2∗ (Ω)

∥∥ |u|p−2u(un − u)
∥∥
L(2∗−1)/2∗ (Ω)

≤ C‖un‖pλ
∥∥ |un|p−2un(un − u)

∥∥
L(2∗−1)/2∗ (Ω)

+ C‖u‖pλ
∥∥ |u|p−2u(un − u)

∥∥
L(2∗−1)/2∗ (Ω)

≤ C
(
‖un‖pλ‖un‖

p−1
Lp(Ω) + ‖u‖pλ‖u‖

p−1
Lp(Ω)

)
‖un − u‖L2∗p/(2∗−p)(Ω) → 0,

(3.7)

as n→∞, where we used Lemma 2.6.
Next, by Lemma 2.6, we know that un → u in Lp(x)(Ω) and Lq(x)(Ω), respec-

tively. Thus,

lim
n→∞

∫
Ω

|un − u|p(x)dx = 0, (3.8)

lim
n→∞

∫
Ω

|un − u|q(x)dx = 0. (3.9)

Finally, it follows from (3.3)-(3.4) and (3.6)-(3.7) that

o(1) = 〈I ′λ(un)− I ′λ(u), un − u〉

= 〈un − u, un − u〉λ +

∫
Ω

(φun |un|p−2un − φu|u|p−2u)(un − u)dx

− α
∫

Ω

(
|un|p(x)−2un − |u|p(x)−2u

)
(un − u)dx

− β
∫

Ω

(
|un|q(x)−2un − |u|q(x)−2u

)
(un − u)dx

= ‖un − u‖λ −
∫

Ω

|un − u|p(x)dx−
∫

Ω

|un − u|q(x)dx,



EJDE-2022/81 SCHRÖDINGER-POISSON SYSTEM WITH VARIABLE EXPONENTS 11

which together with (3.8)-(3.9) implies

lim
n→∞

‖un − u‖λ = 0,

which completes the proof. �

Theorem 3.7. Assume that (A1)–(A5) hold. Then (2.2) has a nontrivial solution
u1
λ in Eλ with Iλ(u1

λ) > 0.

Proof. Thanks to Lemma 3.3, Lemma 3.4 and Theorem 3.2, we deduce that for all
λ > 0 there exists a (PS)cλ sequence {un}n for Iλ on Eλ. In view of Lemma 3.6,
we know that Iλ satisfies (PS)cλ condition, and there exists u1

λ ∈ Eλ such that
I ′λ(u1

λ) = 0 and Iλ(u1
λ) = cλ for all λ > 0. Thus, u1

λ is a solution of (2.2). �

The following proposition plays a fundamental role in giving the second solution
for Problem (1.1).

Proposition 3.8 (Ekeland variational principle, [20, Theorem 1.1]). Let V be a
Banach space and F : V → R∪{+∞} be lower semicontiuous, bounded from below.
Then, for each ε > 0, there exists a point ν ∈ V with

F (ν) ≤ inf
V
F + ε, F (ν) ≤ F (w) + εd(ν, w) for all w ∈ V.

In the next proof, we set Bρ = {u ∈ Eλ : ‖u‖λ < ρ}, where ρ > 0 is given by
Lemma 3.3.

Theorem 3.9. Suppose that (A1)–(A5) hold. Then (1.1) admits another nontrivial
solution u2

λ in Eλ with Iλ(u2
λ) < 0.

Proof. Let us denote c̃λ = infu∈Bρ Iλ, and pick 0 < ε < infu∈∂Bρ Iλ − c̃λ. By

Lemma 3.8, we can choose uε such that

Iλ(uε) ≤ c̃λ + ε, (3.10)

Iλ(uε) ≤ Iλ(u) + ε‖uε − u‖λ (3.11)

for all u ∈ Bρ and u 6= uε. This implies that uε ∈ Bρ because Iλ(uε) ≤ c̃λ + ε <
infu∈∂Bρ Iλ. Let us set

u = uε + τv, ∀v ∈ B1 := {v ∈ Eλ : ‖v‖λ ≤ 1},
where τ > 0 small enough that 0 < τ ≤ ρ− ‖uε‖λ for fixed n large. Then

‖u‖λ = ‖uε + τv‖λ ≤ ‖uε‖λ + τ ≤ ρ,
which implies that u ∈ Bρ. Thus, it follows from (3.11) that

0 ≤ Iλ(uε + τv)− Iλ(uε)

τ
+ ε‖v‖λ,

Therefore, letting τ → 0+, we obtain

〈I ′λ(uε), v〉+ ε‖v‖λ ≥ 0.

By choosing τ < 0 such that |τ | small enough, we use a similar discussion as above
to obtain

−〈I ′λ(uε), v〉+ ε‖v‖λ ≥ 0.

We immediately conclude that
∣∣〈I ′λ(uε), v〉

∣∣ ≤ ε‖v‖λ, for any v ∈ B1. Hence we
know ∣∣〈I ′λ(uε), v〉

∣∣ ≤ ε. (3.12)



12 X. LIN, S. ZHENG EJDE-2022/81

Using (3.10) and (3.12), we can choose a sequence {un} ⊂ Bρ such that

Iλ(un)→ c̃λ, ‖I ′λ(un)‖E∗λ → 0, (3.13)

as n→∞. Thus, {un} is a bounded (PS)c̃λ sequence in the reflexive Banach space
Eλ due to Lemma 3.5. Repeating the process as Lemma 3.6, there exists u2

λ and a
subsequence of {un} still denoted by {un} such that un → u2

λ in Eλ.
Next, we claim that c̃λ < 0. To this end, let us take a nonnegative function

ω0 ∈ Bρ such that ∫
Ω

|ω0|q(x)dx > 0.

Then, for τ ∈ (0, 1) small enough we infer from Lemma 2.8-(2) that

Iλ(τω0)

≤ 1

p
‖τω0‖pλ +

1

2p

∫
Ω

φτω0
|τω0|pdx−

∫
Ω

α

p(x)
|τω0|p(x) +

β

q(x)
|τω0|q(x)dx

≤ 1

p
‖τω0‖pλ +

1

2p
‖τω0‖2pλ −

β

q+

∫
Ω

|τω0|q(x)dx < 0,

(3.14)

where we have used q+ < p < 2p. Thus, we conclude c̃λ < 0. More precisely, it
follows from (3.14) that

c̃λ ≤
1

p
ρp +

1

2p
ρ2p := κ < 0, (3.15)

where constant κ is independent of λ.
In summary, we obtain a nontrivial solution u2

λ of (2.2) satisfying

Iλ(u2
λ) = c̃λ ≤ κ < 0 and ‖u2

λ‖λ < ρ,

which completes the proof. �

Proof of Theorem 1.1. The result follows immediately by combining Theorems 3.7
and 3.9. �

4. Asymptotic behavior of solutions

In this section we study the concentration of solutions for Problem (1.1), which is
stated by Theorem 1.2. Our main idea is motivated by the recent papers [8, 38, 35].

Let Iλ|W 1,p
0 (Ω0) be a restriction of Iλ on W 1,p

0 (Ω0). Note that

Iλ|W 1,p
0 (Ω0) =

1

p
[u]ps,p +

1

p

∫
Ω

|∇u|pdx+
1

2p

∫
Ω

φun |un|pdx

−
∫

Ω

( α

p(x)
|u|p(x) +

β

q(x)
|u|q(x)

)
dx,

By the same method as in the proofs of Lemmas 3.3 and 3.4, we deduce that
Iλ|W 1,p

0 (Ω0) also satisfies mountain pass geometry. Then the (PS)m(Ω0) sequence of

Iλ|W 1,p
0 (Ω0) at the level

m(Ω0) = inf
γ∈Γ̃

max
0≤t≤1

Iλ|W 1,p
0 (Ω0)

(
γ(t)

)
,

can also be constructed, where the set of paths is defined by

Γ̃ = {γ ∈ C1([0, 1];W 1,p
0 (Ω0)) : γ(0) = 0, γ(1) = e}.



EJDE-2022/81 SCHRÖDINGER-POISSON SYSTEM WITH VARIABLE EXPONENTS 13

Clearly, m(Ω0) is independent of λ. Since W 1,p
0 (Ω0) ⊂ Eλ, one has

0 < δ ≤ cλ ≤ m(Ω0) ≤M0 <∞ (4.1)

for all λ > 0.

Proof of Theorem 1.2. For the sequence {λn} with 1 ≤ λn → ∞ as n → ∞, let
uin := uiλn be the critical points of the energy functional Iλn obtained as in Theorem
1.1 for i = 1, 2, that is to say,

I ′λn(u1
n) = 0, Iλn(u1

n) = cλn ,

I ′λn(u2
n) = 0, Iλn(u2

n) = c̃λn .

Thus, from (3.2), (3.15), and (4.1)

Iλn(u2
n) ≤ κ < 0 < δ ≤ Iλn(u1

n) ≤ m(Ω0). (4.2)

By a similar argument as in Lemma 3.5, it is clear that

Iλn(uin) = Iλn(uin)− 1

p−
〈I ′λn(uin), uin〉

≥
(1

p
− 1

p−

)
‖uin‖

p
λ +

( 1

2p
− 1

p−

)∫
Ω

φuin |u
i
n|pdx

− β
( 1

q−
− 1

p−

)∫
Ω

|uin|q(x)dx

≥
(1

p
− 1

p−

)
‖uin‖

p
λ − β

( 1

q−
− 1

p−

)
max

{
Cq

+

q ‖uin‖
q+

λ , Cq
−

q ‖uin‖
q−

λ

}
.

Therefore, we can deduce from (4.2) that there exists a constant C > 0 independent
of λn such that

‖uin‖λn ≤ C,

which shows that {uin}n are uniformly bounded. Passing to a subsequence if nec-

essary, we may assume that there exists ui ∈ W 1,p
0 (Ω0) satisfying uin ⇀ ui in

W 1,p
0 (Ω0). Thanks to Lemma 3.6, we immediately conclude that uin → ui in

Lp(x)(Ω0) and Lq(x)(Ω0), respectively. It follows from Fatou’s lemma that∫
Ω

V (x)|ui|pdx ≤ lim inf
n→∞

∫
Ω

V (x)|uin|pdx ≤ lim inf
n→∞

‖uin‖
p
λn

λn
= 0.

Thus, ui = 0 a.e. in Ω\Ω0, and ui ∈W 1,p
0 (Ω0) because of assumption (A2). Similar

to the proof of Theorem 1.1, we obtain that ui ∈W 1,p
0 (Ω0) for i = 1, 2 are solutions

of Problem (1.5).
Then from (4.2) there exist two positive constants δ, κ independent of λn satis-

fying

I(u2) = lim
n→∞

I(u2
n) ≤ κ < 0 < δ ≤ lim

n→∞
I(u1

n) = I(u1),

which ensures that ui 6= 0 and u1 6= u2. The proof is complete. �



14 X. LIN, S. ZHENG EJDE-2022/81

5. Proof of Theorem 1.3

We are now in a position to consider the existence of infinitely many solutions
of (1.5). The energy functional I : W 1,p

0 (Ω0)→ R associated with (1.5) is

I(u) =
1

p
‖u‖p +

1

2p

∫
Ω0

φu|u|pdx−
∫

Ω0

α

p(x)
|u|p(x)dx−

∫
Ω0

β

q(x)
|u|q(x)dx,

where ‖u‖p := [u]ps,p +
∫

Ω0
|∇u|pdx. Clearly, I ∈ C1(W 1,p

0 (Ω0)) and the critical

points of I are the weak solutions of (1.5).
To achieve the desired results, we need to verify that I satisfies the following

symmetric mountain pass theorem [13, Theorem 2.2].

Lemma 5.1. Let X be an infinite-dimensional Banach space. Suppose that J ∈
C1(X,R) satisfies the (PS)c condition and the following conditions:

(1) J(0) = 0 and J is even;
(2) there exists two constants ρ, δ > 0 satisfying J(u) ≥ δ for all u ∈ X with
‖u‖X = ρ;

(3) for all finite dimensional subspaces Y ⊂ X there exists R = R(Y ) > 0 such
that J(u) ≤ 0 for all u ∈ X \BR(Y ), where BR(Y ) = {u ∈ Y : ‖u‖X ≤ R}.
Then J poses an unbounded sequence of critical values characterized by a
minimax argument.

We first verify that I satisfies Lemma 5.1(3).

Lemma 5.2. Assume that (A3)–(A5) hold. Then, for any finite dimensional

subspace W of W 1,p
0 (Ω0), there exists R0 = R0(W ) such that I(u) < 0 for all

u ∈W 1,p
0 (Ω0) \BR0(W ), where BR0(W ) = {u ∈W : ‖u‖ ≤ R0}.

Proof. LetW be a fixed finite dimensional subspace ofW 1,p
0 (Ω) and R = R(W ) > 1,

for any u ∈W such that ‖u‖ > R. Thus, we have

I(u) =
1

p
‖u‖p +

1

2p

∫
Ω0

φu|u|pdx−
∫

Ω0

α

p(x)
|u|p(x)dx−

∫
Ω0

β

q(x)
|u|q(x)dx

≤ 1

p
‖u‖p +

1

2p
‖u‖2p − 1

p+

∫
Ω0

|u|p(x)dx

≤ 1

p
‖u‖p +

1

2p
‖u‖2p − α

p+
min

{
‖u‖p

−

Lp(x)(Ω0)
, ‖u‖p

+

Lp(x)(Ω0)

}
.

Note that there exists CW > 0 such that ‖u‖Lp(x)(Ω0) ≥ CW ‖u‖, since all norms

are equivalent on finite dimensional Banach space W . Hence, by p+ > p− > 2p, we
obtain

I(u) ≤ 1

p
‖u‖p +

1

2p
‖u‖2p − α

p+
‖u‖p

−
→ −∞, as R→∞.

Therefore, there exists R0 > 0 large enough that I(u) < 0 for all u ∈ W s,p
0 (Ω0),

with ‖u‖ = R and R > R0. Thus the assertion holds. �

Proof of Theorem 1.2. It is obviously to see that I(0) = 0, and I is an even func-
tional. Moreover, the functional I satisfies Lemma 5.1(2) via the proof of Lemma
3.3. Similar to the proof of Lemma 3.6, one can show that I satisfies the (PS)c
condition for any c ∈ R. As a consequence of Lemma 5.1, there exists an unbounded
sequence of solutions for Problem (1.5). �



EJDE-2022/81 SCHRÖDINGER-POISSON SYSTEM WITH VARIABLE EXPONENTS 15

Acknowledgements. We would like to thank the anonymous referees for their
valuable comments and suggestions. This work was supported by the National
Natural Science Foundation of China grant No. 12071021.

References

[1] Albuquerque, F. S.; Carvalho, J. L.; Figueiredo, G. M.; Medeiros, E.; On a planar non-

autonomous Schrödinger-Poisson system involving exponential critical growth, Calc. Var.
Partial Differ. Equ., 60 (2021), Art. 40, 30 pp.

[2] Alves, C. O.; Ferreira, M. C.; Existence of solutions for a class of p(x)-Laplacian equations

involving a concave-convex nonlinearity with critical growth in RN , Topol. Methods Nonlinear
Anal., 45 (2) (2015), 399–422.

[3] Alves, C.O.; Miyagaki, O. H.; Existence and concentration of solution for a class of fractional

elliptic equation in RN via penalization method, Calc. Var. Partial Differ. Equ., 55 (2016),
Art. 47, 19 pp.

[4] Ambrosetti, A.; Brezis, H.; Cerami, G.; Combined effects of concave and convex nonlinearities

in some elliptic problems, J. Funct. Anal., 122 (1994), 519–543.
[5] Ambrosetti, A.; Rabinowitz, P.; Dual variational methods in critical point theorey and ap-

plications, J. Funct. Anal., 14 (1973), 349–381.
[6] Ambrosio, V.; Multiplicity and concentration results for a fractional Schrödinger-Poisson

type equation with magnetic field, Proc. Roy. Soc. Edinburgh Sect. A, 150 (2) (2020), 655–

694.
[7] Bahrouni, A.; Comparison and sub-supersolution principles for the fractional p(x)-Laplacian,

J. Math. Anal. Appl., 458 (2018), 1363–1372.

[8] Bartsch, T.; Pankov, A.; Wang, Z.-Q.; Nonlinear Schrödinger equations with steep potential
well, Commun. Contemp. Math., 3 (4) (2001), 549–569.

[9] Brändle, C.; Colorado, E. E.; de Pablo, A.; Sánchez, U.; A concave-convex elliptic problem

involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh, 143 (2013), 39–71.
[10] Buccheri, S.; da Silva, J. V.; de Miranda, L. H.; A system of local/nonlocal p-Laplacians: the

eigenvalue problem and its asymptotic limit as p → ∞, Asymptot. Anal., 128 (2) (2022),

149–181.
[11] Cencelj, M.; Radulescu, V. D.; Repovs, D.; Double phase problems with variable growth,

Nonlinear Anal., 177 (2018), 270–287.
[12] Chen, Y.; Levine, S.; Rao, M.; Variable exponent, linear growth functionals in image pro-

cessing, SIAM J. Appl. Math., 66 (2006), 1383–1406.

[13] Colasuonno, F.; Pucci, P.; Multiplicity of solutions for p(x)-polyharmonic elliptic Kirchhoff
equations, Nonlinear Anal., 74 (2011), 5962–5974.

[14] Crespo-Blanco, Á.; Gasiński, L.; Harjulehto, P.; Winkert, P.; A new class of double phase
variable exponent problems: existence and uniqueness, J. Differential Equ., 323 (2022), 182–

228.
[15] Di Nezza, E.; Palatucci, G.; Valdinoci, E.; Hitchhiker’s guide to the fractional Sobolev spaces,

Bull. Sci. Math., 136 (5) (2012), 521–573.

[16] Dipierro, S.; Proietti Lippi, E.; Valdinoci, E.; Linear theory for a mixed operator with Neu-

mann conditions, Asymptot. Anal., 128 (4) (2022), 571–594.
[17] Dipierro, S.; Valdinoci, E.; Description of an ecological niche for a mixed local/nonlocal

dispersal: an evolution equation and a new Neumann condition arising from the superposition
of Brownian and Lévy processes, Phys. A, 575 (2021), Art. 126052, 20 pp.

[18] Du, M.; Tian, L. X.; Wang, J.; Zhang, F. B.; Existence and asymptotic behavior of solutions

for nonlinear Schrödinger-Poisson systems with steep potential well, J. Math. Phys., 57 (3)
(2016), 031502, 19 pp.

[19] Du, Y.; Su, J. B.; Wang, C.; On a quasilinear Schrödinger-Poisson system, J. Math. Anal.

Appl., 505 (1) (2022), Art. 125446, 14 pp.
[20] Ekeland, I.; Nonconvex minimization problems, Bull. Am. Math. Soc., 1 (1979), 443–473.

[21] Figueiredo, G. M.; Siciliano, G.; Existence and asymptotic behaviour of solutions for a quasi-

linear Schrödinger-Poisson system with a critical nonlinearity, Z. Angew. Math. Phys., 71
(2020), Art. 130, 21 pp.



16 X. LIN, S. ZHENG EJDE-2022/81

[22] Ho, K.; Sim, I.; Existence and muliplicity of solutions for degenerate p(x)-Laplace equations

involving concave-convex type nonlinearities with two parameters, Taiwanese J. Math., 19

(2015), 1469–1493.
[23] Jeanjean, L.; Le, T. T.; Multiple normalized solutions for a Sobolev critical Schrödinger-

Poisson-Slater equation, J. Differential Equ., 303 (2021), 277–325.

[24] Kim, I. H.; Kim, Y.-H.; Oh, M. W.; Zeng, S. d.; Existence and multiplicity of solutions
to concave-convex-type double-phase problems with variable exponent, Nonlinear Anal. Real

World Appl., 67 (2022), Art 103627, 25 pp.

[25] Lin, X. L.; Zheng, S. Z.; Multiplicity and asymptotic behavior of solutions to fractional
(p, q)-Kirchhoff type problems with critical Sobolev-Hardy exponent, Electron. J. Differential

Equations, 2021(66) (2021), 20 pp.

[26] Lv, H. L., Zheng, S. Z.; Feng, Z. S.; Existence results for nonlinear Schrödinger equations
involving the fractional (p, q)-Laplacian and critical nonlinearities, Electron. J. Differential

Equations, 2021(100) (2021), 24 pp.
[27] Pucci, P.; Temperini, L.; Existence for fractional (p, q) systems with critical and Hardy terms

in RN , Nonlinear Anal., 211 (2021), Art. 112477, 33 pp.

[28] Rǎdulescu, V. D.; Repovš, D. D.; Partial Differential Equations with Variable Exponents,
CRC Press, Boca Raton, FL, 2015.

[29] Ružička, M.; Electrorheological Fluids: Modeling and Mathematical Theory, Springer-Verlag,

Berlin, 2002.
[30] Sun, M. Z.; Su, J. B.; Zhao, L. G.; Infinitely many solutions for a Schrödinger-Poisson

system with concave and convex nonlinearities, Discrete Contin. Dyn. Syst., 35 (1) (2015),

427–440.
[31] Sun, J. T.; Wu, T.-F.; On Schrödinger-Poisson systems involving concave-convex nonlineari-

ties via a novel constraint approach, Commun. Contemp. Math., 23 (6) (2021), Art. 2050048,

25 pp.
[32] Vaira, G.; Ground states for Schrödinger-Poisson type systems, Ric. Mat., 2 (2011) 263–297.

[33] Weiss, C. J.; van Bloemen Waanders, B. G.; Antil, H.; Fractional operators applied to geo-
physical electromagnetics, Geophys. J. Intern., 220 (2) (2020), 1242–1259.

[34] Willem, M.; Minimax Theorems, Progress in Nonlinear Differential Equations and their

Applications, 24. Birkhäuser Boston, Inc., Boston, 1996.
[35] Xiang, M. Q.; Zhang, B. L.; Yang, D.; Multiplicity results for variable-order fractional Lapla-

cian equations with variable growth, Nonlinear Anal., 178 (2019), 190–204.

[36] Yang, Z.-L.; Ou, Z.-Q.; Nodal solutions for Schrödinger-Poisson systems with concave-convex
nonlinearities, J. Math. Anal. Appl., 499 (1) (2021), Art. 125006, 15 pp.

[37] Zeng, S. D.; Rǎdulescu, V. D.; Winkert, P.; Double phase obstacle problems with variable

exponent, Adv. Differential Equ., 27(9-10) (2022), 611–645.
[38] Zhang, F.; Du, M.; Existence and asymptotic behavior of positive solutions for Kirchhoff type

problems with steep potential well, J. Differ. Equ., 269 (11) (2020), 10085–10106.

Xiaolu Lin
Department of Mathematics, Beijing Jiaotong University, Beijing 100044, China

Email address: 19118003@bjtu.edu.cn

Shenzhou Zheng (corresponding author)
Department of Mathematics, Beijing Jiaotong University, Beijing 100044, China

Email address: shzhzheng@bjtu.edu.cn


	1. Introduction
	2. Preliminaries
	3. Proof of Theorem 1.1
	4. Asymptotic behavior of solutions 
	5. Proof of Theorem 1.3
	Acknowledgements

	References