Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 93, pp. 1–13.

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

DOI: 10.58997/ejde.2025.93

MIXED LOCAL AND NONLOCAL CRITICAL

SCHRÖDINGER-KIRCHHOFF-POISSON TYPE SYSTEMS WITH

LOGARITHMIC PERTURBATION

SHENGBING DENG, GUORONG ZENG

Abstract. In this article, we consider the mixed local and nonlocal critical Schrödinger-Kirchhoff-
Poisson type system with logarithmic perturbation

−M(

∫
Ω
|∇u|2 dx)∆u+ a(−∆)su+ λϕu = η|u|q−2u ln |u|2 + |u|4u, in Ω,

−∆ϕ = u2, in Ω,

ϕ = u = 0, in R3 \ Ω.

where Ω ⊂ R3 is a bounded domain with smooth boundary, 0 < s < 1, 4 < q < 6, λ, η > 0 are

two parameters, M(t) = a+ bt and a, b are nonnegative constants. With the help of variational
methods, the existence of a non-trivial ground state solution is obtained.

1. Introduction and statement of main result

In this article, we consider the mixed local and nonlocal critical Schrödinger-Kirchhoff-Poisson
type system with logarithmic perturbation

−M
(∫

Ω

|∇u|2 dx
)

∆u+ a(−∆)su+ λϕu = η|u|q−2u ln |u|2 + |u|4u, in Ω,

−∆ϕ = u2, in Ω,

ϕ = u = 0, in R3 \ Ω.

(1.1)

where Ω ⊂ R3 is a bounded domain with smooth boundary ∂Ω, 0 < s < 1, 4 < q < 6, λ, η > 0 are
two parameters, M(t) = a+ bt and a, b are nonnegative constants.

When a = 1 and b = 0, problem (1.1) reduces to the boundary value problem

−∆u+ (−∆)su+ λϕu = f(x, u), in Ω,

−∆ϕ = u2, in Ω,

ϕ = u = 0, in RN \ Ω.

(1.2)

We emphasize that operators of the form −∆ + (−∆)s are referred to as mixed operators, com-
bining both local and nonlocal characteristics as well as different orders of differentiation, and
this operators, derived from the superposition of the classical Laplacian −∆ and the fractional
Laplacian (−∆)s for a fixed parameter s ∈ (0, 1) which, up to a normalization constant, is defined
by

(−∆)su(x) = lim
ϵ→0+

∫
RN\Bϵ(x)

u(x) − u(y)

|x− y|N+2s
dy.

2020 Mathematics Subject Classification. 35M12, 35R11, 35A15, 35B33.
Key words and phrases. Ground state solution; mixed local-nonlocal operators; logarithmic nonlinearity;

Schrödinger-Kirchhoff-Poisson system.
©2025. This work is licensed under a CC BY 4.0 license.
Submitted August 8, 2025. Published October 6, 2025.

1



2 S. DENG, G. ZENG EJDE-2025/93

As we know, the operators from the superposition of two stochastic processes with different
scales, such as a classical random walk and a Lévy flight. Furthermore, the investigation of
mixed operators is a very topical subject of investigation, arising naturally in several fields. For
more information, one can see [13, 22, 23] and the references therein. The mixed operators have
attracted the attention of many mathematicians. From the viewpoint of mathematics, there is a
lack of scale invariances, which may produce unexpected complexity. More specifically, Lin et al.
[19] concerned the mixed local and nonlocal Poisson system

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

−∆ϕ = up, in Ω,

ϕ = u = 0, in RN \ Ω.

here N ≥ p, Ω ⊂ RN is a bounded domain, V (x) is a potential function, p(x), q(x) are variable
exponents. They investigated the multiplicity of solutions for λ > 0 and the concentration and
multiplicity of solutions for λ → ∞. Biagi et al. [7] showed the mixed Sobolev inequality and
investigated the optimal constant. In [4, 6, 8, 25], the authors proved strong maximum principle,
regularity theory, eigenvalues theory and existence results.

In (1.2), if s = 1, then system (1.2) is reduced to the classical Laplace Schrödinger-Poisson
system, which has been widely studied. It is widely known that the existence of solutions for
system (1.2) can be studied using variational methods, provided that appropriate assumptions are
made. For instance, one can see [1, 3, 2, 17, 21, 24] and the references therein.

Then, considering solely the first equation in (1.1) with the potential set to zero, we obtain the
problem

−(a+ b

∫
Ω

u2 dx)∆u+ a(−∆)su = f(x, u), in Ω,

u = 0, in RN \ Ω.

(1.3)

which was proposed by Kirchhoff (see [11]) as a generalization of well-know D’Alembert wave
equation

∂2u

∂t2
−
(P0

h
+

E

2L

∫ L

0

|∂u
∂x

|dx
)∂2u
∂x2

= 0.

For mixed operators involving the Kirchhoff function, only Wang et al. [31] considered equation

(1.3) with f(x, u) =
( ∫

Ω
|u(y)|p
|x−y|µ dy

)
|u|p−2u+ λ|u|q−2u, 1 < q < 2 < 2p, 0 < µ < N , employing the

non-smooth variational principle to prove the existence of solutions. For s = 1, i.e. the system
reduces to the Laplace operator with Kirchhoff function, which has been extensively investigated
by many researchers using variational methods, see [14, 20, 28, 18]. Moreover, some studies of
Schrödinger-Kirchhoff-Poisson type system with the Laplace operator can be referred to [26, 30]
and the references therein.

In recent years, equations with logarithmic nonlinearity have attracted increasing interest,
mainly due to their wide application in modeling various phenomena in the physical sciences.
The main challenge presented by logarithmic nonlinearity arises from its characteristic of sign
change, which means it does not conform to the standard of monotonicity, nor does it follow the
Ambrosetti-Rabinowitz conditions. Therefore, studying problems involving logarithmic nonlinear-
ity presents a challenge. In the past few years, a significant amount of research work has focused
on this area. For example, Deng et al. [12] considered the equation

−∆u = λu+ µu lnu2 + |u|2
∗−2u, in Ω,

u = 0, in ∂Ω,

where 2∗ := 2N
N−2 is the critical Sobolev exponent. By using Mountain Pass Lemma they showed

that the problem admits at least one nontrivial weak solution under some appropriate assumptions



EJDE-2025/93 SCHRÖDINGER-KIRCHHOFF-POISSON TYPE SYSTEMS 3

on λ and µ. Li et al. [18] studied the critical Kirchhoff problems with logarithmic nonlinearity,

−(a+ b

∫
Ω

|∇u|2 dx)∆u = λ|u|q−2u ln |u|2 + |u|4u in Ω,

u = 0, in ∂Ω,
(1.4)

where Ω ⊂ R3 is a bounded domain with smooth boundary ∂Ω, 4 < q < 6, λ > 0 is a parameter,
a ≥ 0, b > 0 and they investigated the existence of a solution. Additionally, many fascinating
studies have been performed on problems involving logarithmic nonlinearity. Among the wide
range of such studies, we refer readers interested to [27, 32, 33] and the related references for
further exploration.

Motivated by the results mentioned above, the aim of this paper is to consider the existence
of nontrivial ground state solution for problem (1.1). Under some natural assumptions, by using
the concentration-compactness principle and the mountain pass theorem, we obtain the existence
result of nontrivial ground state solution. To the best of our knowledge, there is little literature
which essentially attacks the Brezis-Nirenberg problem for mixed local and nonlocal operators
critical Schrödinger-Kirchhoff-Poisson type system with logarithmic nonlinearity. Now, we are
ready to state the main result of this paper.

Theorem 1.1. Assume that a > 0. If q ∈ (4, 6), then problem (1.1) has at least one nontrivial
ground state solution provided that either

(1) s ≤ 1
2 and λ, η > 0, or

(2) s > 1
2 , λ ∈ (0, λ∗) and η > η∗.

Corollary 1.2. Assume that a = 0. If q ∈ (4, 6), then problem (1.1) reduces to the Laplace
degenerate Schrödinger-Kirchhoff-Poisson type system, which admits at least one nontrivial ground
state solution for all λ, η > 0.

Corollary 1.3. Assume that b = 0. If q ∈ (4, 6), then problem (1.1) reduces to the critical
Schrödinger-Poisson type system, which admits at least one nontrivial ground state solution pro-
vided that either

(1) s ≤ 1/2 and λ, η > 0, or

(2) s > 1/2, λ ∈ (0, λ̂∗) and η > η̂∗.

To prove Theorem 1.1, which is the critical case, we face a complication due to the non-
compactness of the embedding of X0(Ω) ↪→ L6(Ω), the standard variational techniques do not ap-
ply in a straightforward way. The lack of compactness of this problem is expressed by the fact that
the functional does not satisfy the Palais-Smale condition in all the energy range (−∞,+∞). To
surmount this challenge, we attempt to recover compactness through the concentration-compactness
principle. Moreover, there is a Kirchhoff term in (1.1), which will affect the functional satisfies
the Palais-Smale condition in some range.

The rest of this article is organized as follows. In Section 2, we introduce the functional setting
of (1.1) and some necessary definitions and preliminary lemmas are given. In Section 3, we prove
Theorem 1.1.

2. Preliminaries and variational setting

We denote by | · |p the usual Lp-norm. Let s ∈ (0, 1), the fractional Sobolev space is

Hs(Ω) =
{
u ∈ L2(Ω) :

∫
R3

∫
R3

|u(x) − u(y)|2

|x− y|3+2s
dx dy < +∞

}
,

which is endowed with the semi-norm

[u]s :=
(∫

R3

∫
R3

|u(x) − u(y)|2

|x− y|2s+3
dx dy

)1/2

which is called the Gagliardo seminorm of u. The Sobolev space H1(Ω) is defined as the Banach
space of weakly differentiable functions u : Ω → R equipped with the following norm

∥u∥2H1(Ω) = |u|22 + |∇u|22.



4 S. DENG, G. ZENG EJDE-2025/93

The space H1(R3) is defined similarly.
We define the homogeneous Sobolev space H1

0 (Ω) as the completion of the space C∞
0 (Ω) under

the norm ∥u∥ = |∇u|2. However, to address the boundary condition specified in equation (1.1), it
becomes necessary to extend our consideration to functions that are defined over the entire space
R3. For this, H1

0 (Ω) is not enough and we consider the functional space

X0(Ω) =
{
u ∈ H1(R3) : u|Ω ∈ H1

0 (Ω) and u = 0 a.e. in R3 \ Ω
}

endowed with the norm

∥u∥X0
=

(
|∇u|22 + [u]2s

)1/2

.

It is well known that X0(Ω) is a real separable and uniformly convex space. Also, we have the
following relation between the gradient seminorm and the Gagliardo norm.

Lemma 2.1 ([9, Lemma 2.1]). Let Ω ⊂ R3 be bounded domain, there exists a constant C =
C(s) ≥ 1, such that

[u]s ≤ C∥u∥H1(Ω), ∀u ∈ H1(Ω).

Lemma 2.2 ([10, Proposition 2.2]). Under the same hypothesis of Lemma 2.1, there exists a
constant C = C(s,Ω) ≥ 1, such that∫

R3

∫
R3

|u(x) − u(y)|2

|x− y|3+2s
dx dy ≤ C

∫
Ω

|∇u|2 dx, ∀u ∈ H1
0 (Ω)

From Lemmas 2.1 and 2.2, it is easy to know that norms ∥u∥ and ∥u∥X0
are equivalent, when

u ∈ X0(Ω). Since Ω is a bounded domain, we have H1
0 (Ω) ↪→  Lp(Ω) continuously for p ∈ [1, 6] and

compactly for p ∈ [1, 6). Naturally, we define Sn as the best Sobolev constant for the embedding
H1

0 (Ω) ↪→  L6(Ω), then

Sn = inf
u∈H1

0 (R3)\{0}

∫
R3 |∇u|2 dx

(
∫
R3 |u|6 dx)1/3

.

Because the norms ∥u∥ and ∥u∥X0
are equivalent, we can obtain X0(Ω) ↪→  Lp(Ω) continuously

for p ∈ [1, 6] and compactly for p ∈ [1, 6). Moreover, we have the following result.

Lemma 2.3 ([7, Theorem 1.1]). Let Sn,s as the best Sobolev constant for the embedding X0(Ω) ↪→
 L6(Ω). Then

Sn,s(Ω) = inf
u∈X0(Ω)\{0}

∥u∥2X0

|u|26
= Sn.

Definition 2.4. We say that u ∈ X1
0 (Ω) is a weak solution of problem (1.1), if

a

∫
Ω

∇u · ∇v + b∥u∥2
∫
Ω

∇u · ∇v + a

∫
R3

∫
R3

(u(x) − u(y))(v(x) − v(y))

|x− y|3+2s
dx dy

+ λ

∫
Ω

ϕuv dx

= η

∫
Ω

|u|q−2uv ln |u|2 dx+

∫
Ω

|u|4uv dx

(2.1)

for all v ∈ X1
0 (Ω).

The following result is well known.

Lemma 2.5 (see [1, 21]). For each u ∈ X0(Ω), there exists a unique element ϕu ∈ X0(Ω) such
that −∆ϕu = u2, moreover, ϕu has the following properties:

(a) there exists c > 0 such that ∥ϕu∥ ≤ c∥u∥2 and∫
Ω

|∇ϕu|2 dx =

∫
Ω

ϕuu
2 dx ≤ c∥u∥4; (2.2)

(b) ϕu ≥ 0 and ϕtu = t2ϕu, ∀t > 0;



EJDE-2025/93 SCHRÖDINGER-KIRCHHOFF-POISSON TYPE SYSTEMS 5

(c) if un ⇀ u in X0(Ω), then ϕun
⇀ ϕu in X0(Ω) and

lim
n→+∞

∫
Ω

ϕun
u2n dx =

∫
Ω

ϕuu
2 dx. (2.3)

By the above lemma, (u, ϕ) ∈ X0(Ω) ×H1
0 (Ω) is a solution of (1.1) if and only if ϕ = ϕu and

u ∈ X0(Ω) is a solution of the problem

−M
(∫

Ω

|∇u|2 dx
)

∆u+ a(−∆)su+ λϕuu = η|u|q−2u ln |u|2 + |u|4u, in Ω,

u = 0, in R3 \ Ω.

We define the energy functional associated with problem (1.1) by

I(u) =
a

2
∥u∥2X0

+
b

4
∥u∥4 +

λ

4

∫
Ω

ϕu2 dx+
2η

q2
|u|qq −

η

q

∫
Ω

|u|q ln |u|2 dx− 1

6
|u|66. (2.4)

Since 4 < q < 6, it is easy to prove that the functional I(u) is well defined and is a C1 functional
in X0(Ω). Naturally, u is a weak solution to problem (1.1) if and only if u is a critical point of I.

3. Proof of main result

Firstly, we prove that I(u) possesses a kind of mountain-pass geometrical structure.

Theorem 3.1 (see [29]). Let X be a real Banach space and J ∈ C1(X;R) with J(0) = 0. Suppose
that

(i) there exist ρ, α > 0 such that J(u) ≥ α for all u ∈ X, with∥u∥X = ρ;
(ii) there exists e ∈ X satisfying ∥e∥X > ρ such that J(e) < 0.

Define Γ = {γ ∈ C1
(

[0, 1];X
)

: γ(0) = 1, γ(1) = e}. Then

c = inf
γ∈Γ

max
0≤t≤1

J(γ(t)) ≥ α

and there exists a (PS)c sequence (un)n ⊆ X.

To use Theorem 3.1, we show that the functional E satisfies the mountain pass geometry (i)
and (ii).

Lemma 3.2. The functional I has a mountain pass geometry shown as:

(i) there exist α, ρ > 0 such that I(u) ≥ α for ∥u∥X0
= ρ;

(ii) there exists ω ∈W 1,p
0 (Ω) such that I(ω) < 0.

Proof. Using the well-know vector inequality

|tq ln t| ≤ 1

eq
, ∀t ∈ (0, 1), tq ln t ≤ 1

eδ
tq+δ, ∀t ≥ 1, (3.1)

where δ > 0 is chosen to satisfy q + δ < 6, we obtain |u|q ln |u|2 ≤ 2
eq + 2

eδ |u|
q+δ.

By applying (3.1) with q + δ < 6 and using the Sobolev embedding inequality, one has

I(u) =
a

2
∥u∥2X0

+
b

4
∥u∥4 +

λ

4

∫
Ω

ϕuu
2 dx+

2η

q2
|u|qq −

η

q

∫
Ω

|u|q ln |u|2 dx− 1

6
|u|66

≥ a

2
∥u∥2X0

− η

q

∫
Ω

|u|q ln |u|2 dx− 1

6
|u|66

≥ a

2
∥u∥2X0

− C∥u∥q+δ
X0

− 1

6
S−3
n ∥u∥6X0

= ∥u∥2X0
(
a

2
− C∥u∥q+δ−2

X0
− 1

6
S−3
n ∥u∥4X0

).

Then there exist positive constants α, ρ such that

I(u) ≤ α, for all ∥u∥X0
= ρ.

From this, (i) is proved.



6 S. DENG, G. ZENG EJDE-2025/93

To prove (ii), we fix v ∈W 1,p
0 (Ω) \ {0} and t > 0. According to (2.2), we have

I(tv) =
at2

2
∥v∥2X0

+
bt4

4
∥v∥4 +

λt4

4

∫
Ω

ϕvv
2 dx+

2ηtq

q2
|v|qq −

ηtq

q

∫
Ω

|v|q ln |tv|2 dx− t6

6
|v|66

≤ at2

2
∥v∥2X0

+
bt4

4
∥v∥4 +

λt4

4
c∥u∥4 +

2ηtq

q2
|u|qq −

t6

6
|v|66,

From q < 6, we deduce that there exists t1 > 0 large enough such that ∥t1v∥X0
> ρ and I(t1v) < 0.

Taking ω = t1v, (ii) also holds. □

As the functional I(u) does not meet the (PS)c condition for every value of c, we will restrict
c to a range where the (PS)c condition holds true. The primary method employed for this is the
concentration-compactness principle of Lions (see [15, 16]).

Lemma 3.3. Let {un} ⊂ X0(Ω) be a Palais-Smale sequence for I(u) at the level c with c < c(A),

where c(A) := a
2A+ b

4A
2 − 1

6
A3

S3
n

with A =
bS3

n+
√

b2S6
n+4aS3

n

2 , that is I(un) → c and I ′(un) → 0 as

n → ∞. Then there exist a subsequence of {un} and a u ∈ X0(Ω) such that un → u in X0(Ω) as
n→ ∞.

Proof. First, we claim that {un} is bounded. From I(un) → c and I ′(un) → 0 as n → ∞, we
obtain, for n large enough, that

c+ 1 + o(1)∥un∥

≥ I(un) − 1

q
⟨I ′(un), un⟩

= (
1

2
− 1

q
)a∥un∥2X0

+ (
1

4
− 1

q
)(b∥un∥4 + λ

∫
Ω

ϕun
u2n dx) +

2η

q2
|un|qq + (

1

q
− 1

6
)|un|66

≥ (
1

2
− 1

q
)a∥u∥2 + (

1

4
− 1

q
)b∥un∥4,

according to the assumption that a+ b > 0 and 4 < q < 6, which implies that {un} is bounded in
X0(Ω). If ∥un∥ → 0 as n→ ∞, the proof is complete. Thus, we assume that ∥un∥ ̸→ 0 as n→ ∞,
up to a subsequence, we may assume that

un ⇀ u weakly in X0(Ω),

un → u strongly in Lp(Ω)(1 ≤ p < 6),

un → u a.e. in Ω,

|∇un|2 ⇀ µ and |un|6 ⇀ ν,

where µ and ν are a nonnegative bounded measures on Ω. Then, by concentration compactness
principle, there exists some at most countable index set J such that

ν = |u|6 +
∑
j∈J

νjδxj , νj > 0, (3.2)

µ ≥ |∇u|2 +
∑
j∈J

µjδxj
, µj > 0, (3.3)

Snν
1/3
j ≤ µj , (3.4)

where δxj is the Dirac measure mass at xj ∈ Ω.
Take ψ(x) ∈ C∞

0 (Ω) such that 0 ≤ ψ ≤ 1,

ψ(x) =

{
1, if x ∈ B(xj , ρ)

0, if x ∈ Ω \B(xj , 2ρ)

and |∇ψ|∞ ≤ 2.



EJDE-2025/93 SCHRÖDINGER-KIRCHHOFF-POISSON TYPE SYSTEMS 7

For ρ > 0, we define ψj
ρ = ψ(

x−xj

ρ ), where j ∈ J . Since I ′(un) → 0 and (ψj
ρun)n is bounded,

⟨I ′(un), ψj
ρun⟩ → 0 as n→ ∞, that is,

(a+ b∥u∥2)

∫
Ω

|∇un|2ψj
ρ dx+ a

∫
R3

∫
R3

|un(x) − un(y)|2ψj
ρ

|x− y|3+2s
dx dy + λ

∫
Ω

ϕun
u2nψ

j
ρ dx

= −(a+ b∥u∥2)

∫
Ω

un∇un∇ψj
ρ dx−G2(un, ψ

j
ρun)

+ η

∫
Ω

|un|qψj
ρ ln |un|2 dx+

∫
Ω

|un|6ψj
ρ dx+ on(1),

(3.5)

where

G2(un, ψ
j
ρun) = a

∫
R3

∫
R3

(un(x) − un(y))un(x)(ψj
ρ(x) − ψj

ρ(y))

|x− y|3+2s
dx dy.

It is easy to prove that

lim
ρ→0

lim
n→∞

∫
R3

∫
R3

|un(x) − un(y)|2ψj
ρ

|x− y|3+2s
dx dy > 0. (3.6)

By (2.3), we have

lim
ρ→0

lim
n→∞

∫
Ω

ϕunu
2
nψ

j
ρ dx = lim

ρ→0

∫
B(xj ,2ρ)

ϕuu
2ψj

ρ dx = 0. (3.7)

Note that the Hölder inequality implies

|G2(un, ψ
j
ρun)|

≤
∫
R3

∫
R3

|un(x) − un(y)||un(x)||ψj
ρ(x) − ψj

ρ(y)|
|x− y|3+2s

dx dy

≤
(∫

R3

∫
R3

|un(x) − un(y)|2

|x− y|3+2s
dx dy

)1/2(∫
R3

∫
R3

|un(x)|2|ψj
ρ(x) − ψj

ρ(y)|2

|x− y|3+2s
dx dy

)1/2

≤ C
(∫

R3

∫
R3

|un(x)|2|ψj
ρ(x) − ψj

ρ(y)|2

|x− y|3+2s

)1/2

.

With the same argument as in the proof of [34, Lemma 3.4], we have

lim
ρ→0

lim
n→∞

∫
R3

∫
R3

|un(x)|2|ψj
ρ(x) − ψj

ρ(y)|2

|x− y|3+2s
= 0. (3.8)

It follows that limρ→0 limn→∞ |G2(un, ψ
j
ρun)| = 0. Using the Hölder inequality again, we have

lim
n→∞

∫
Ω

un∇un∇ψj
ρ dx ≤ lim

n→∞
C
(∫

Ω

|un∇ψj
ρ|2 dx

)1/2

≤ C
(∫

B(xj ,2ρ)

|u|2|∇ψj
ρ|2 dx

)1/2

.

(3.9)

so we can get limρ→0 limn→∞
∫
Ω
un∇un∇ψj

ρ dx = 0.
On the other hand, since un → u a.e. in Ω as n→ ∞, we obtain

ψj
ρ|un|q ln |un|2 → ψj

ρ|u|q ln |u|2, a.e. in Ω as n→ ∞, (3.10)

With the help of (3.1), (3.10) and Lebesgue’s dominated convergence theorem, we obtain

lim
n→∞

∫
Ω

ψj
ρ|un|q ln |un|2 dx =

∫
Ω

ψj
ρ|u|q ln |u|2 dx,

which then guarantees that

lim
ρ→0

lim
n→∞

∫
Ω

ψj
ρ|un|q ln |un|2 dx = lim

ρ→0

∫
B(xj ,2ρ)

ψj
ρ|u|q ln |u|2 dx = 0. (3.11)



8 S. DENG, G. ZENG EJDE-2025/93

Since ψj
ρ has compact support, letting n → ∞, ρ → 0 in (3.5), we deduce from (3.3), (3.6)-(3.9)

and (3.11) that

(a+ bµj)µj ≤ νj . (3.12)

Next, we prove that µj = νj = 0. Otherwise, combining (3.12) with (3.4), we have

µ2
j − bS3

nµj − aS3
n ≥ 0 and Sn(ν

1/3
j )2 − bS3

nν
1/3
j − aS2

n ≥ 0,

which yields

µj ≥ A, νj ≥
A3

S3
n

, (3.13)

where A :=
bS3

n+
√

b2S6
n+4aS3

n

2 satisfying A2 − bS3
nA − aS3

n = 0. Then, from I(un) → c and
I ′(un) → 0 as n → ∞, letting n → ∞, ρ → 0 on both sides of the above inequality and in view
of (3.13), we have

c = I(un) − 1

q
⟨I ′(un), un⟩ + o(1)

= (
1

2
− 1

q
)a∥un∥2X0

+ (
1

4
− 1

q
)(b∥un∥4 + λ

∫
Ω

ϕun
u2n dx) +

2η

q2
|un|qq + (

1

q
− 1

6
)|un|66 + o(1)

≥ (
1

2
− 1

q
)a∥un∥2 + (

1

4
− 1

q
)b∥un∥4 + (

1

q
− 1

6
)|un|66 + o(1)

≥ (
1

2
− 1

q
)aµj + (

1

4
− 1

q
)bµ2

j + (
1

q
− 1

6
)νj

≥ (
1

2
− 1

q
)aA+ (

1

4
− 1

q
)bA2 + (

1

q
− 1

6
)
A3

S3
n

=
a

2
A+

b

4
A2 − 1

6

A3

S3
n

,

where we use that fact in the above equality that A2 − bS3
nA − aS3

n = 0. This contradicts the
assumption that c < c(A). Thus, µj = νj = 0 and we obtain

lim
n→∞

∫
Ω

|un|6 dx =

∫
Ω

|u|6 dx. (3.14)

Now, we are ready to show that {un} converges strongly to u in X0(Ω) as n → ∞. Let us write
vn = un − u, hence, by the Brezis-Lieb Lemma, we have

∥un∥2 = ∥vn∥2 + ∥u∥2 + o(1),

∥un∥4 = ∥vn∥4 + ∥u∥4 + 2∥vn∥2∥u∥2 + o(1),

[un]2s = [vn]2s + [u]2s + o(1),

(3.15)

by (2.3) and (3.11), we obtain ∫
Ω

ϕunu
2
n dx =

∫
Ω

ϕuu
2 dx+ o(1),∫

Ω

|un|q ln |un|2 dx =

∫
Ω

|u|q ln |u|2 dx+ o(1).

(3.16)

If vn = un − u and [vn]s ≤ ∥vn∥ → 0, the proof is complete. Otherwise there exists a subse-
quence (still denoted by vn) such that limn→∞[vn]s = l, limn→∞ ∥vn∥ = k, where l, k are positive
constants.

Thus, from (3.15), (3.16) and I ′(un) → 0, it follows that

o(1) = ⟨I ′(un), u⟩

= a∥u∥2X0
+ b∥un∥2∥u∥2 + λ

∫
Ω

ϕuu
2 dx− η

∫
Ω

|u|q ln |u|2 dx−
∫
Ω

|u|6 dx+ o(1)

= a∥u∥2X0
+ bk2∥u∥2 + b∥u∥4 + λ

∫
Ω

ϕuu
2 dx− η

∫
Ω

|u|q ln |u|2 dx−
∫
Ω

|u|6 dx+ o(1).

(3.17)



EJDE-2025/93 SCHRÖDINGER-KIRCHHOFF-POISSON TYPE SYSTEMS 9

By (3.14), (3.15), (3.17) and recalling that {un} is bounded in H1
0 (Ω), we obtain

o(1) = ⟨I ′(un), un⟩

= a∥un∥2X0
+ b∥un∥4 + λ

∫
Ω

ϕuu
2 dx− η

∫
Ω

|u|q ln |u|2 dx−
∫
Ω

|u|6 dx

= a∥u∥2X0
+ a∥vn∥2X0

+ b∥u∥4 + b∥vn∥4 + 2b∥vn∥2∥u∥2

+ λ

∫
Ω

ϕuu
2 dx− η

∫
Ω

|u|q ln |u|2 dx−
∫
Ω

|u|6 dx+ o(1)

= ⟨I ′(un), u⟩ + a∥vn∥2X0
+ b∥vn∥4 + b∥vn∥2∥u∥2 + o(1)

= ak2 + al2 + bk4 + bk2∥u∥2 + o(1),

which implies that k = l = 0, so un converges strongly to u in X0(Ω). The proof is complete. □

According to above Lemmas, we know that there exists a sequence {un}n ∈ X0(Ω) such that
I(un) → c and I ′(un) → 0, where

c = inf
γ∈Γ

max
0≤t≤1

I(γ(t)) with Γ = {γ ∈ C1
(

[0, 1];X
)

: γ(0) = 1, γ(1) = e}.

To better align with our aim, as illustrated in reference [29], we can employ the following equivalent
definition of c, which is provided by

c = inf
u∈N

I(u) = inf
u∈X0(Ω)\{0}

max
t≥0

I(tu),

where the Nehari manifold N associated to I is defined by

N = {u ∈ X0(Ω) \ {0} : ⟨I ′(u), u⟩ = 0}.

Fixed u ∈ X0(Ω) and define the function of the form Ju : t → I(tu) for t > 0, such map is
famous in bifurcation theory. And the maps are closely related to the Nehari manifold defined by

N = {u ∈ X0(Ω) : J ′
u(1) = 0}.

We split N into three parts:

N+ = {u ∈ N : J ′′
u (1) > 0},

N− = {u ∈ N : J ′′
u (1) < 0},

N 0 = {u ∈ N : J ′′
u (1) = 0}.

which corresponds to local minima, local maxima and points of inflexion of the fibering maps. It
is well known that each nontrival solution to problem (1.1) belongs to N . Furthermore, we have
the following lemma.

Lemma 3.4. N = N−

Proof. For each fixed u ∈ N , we have

J ′
u(1) = a∥u∥2X0

+ b∥u∥4 + λ

∫
Ω

ϕuu
2 dx− η

∫
Ω

|u|q ln |u|2 dx− |u|66,

and

J ′′
u (1) = a∥u∥2X0

+ 3b∥u∥4 + 3λ

∫
Ω

ϕuu
2 dx− 2η|u|qq − (q − 1)η

∫
Ω

|u|q ln |u|2 dx− 5|u|66

= −
[
(q − 2)a∥u∥2X0

+ (q − 4)b∥u∥4 + (q − 4)λ

∫
Ω

ϕuu
2 dx+ 2η|u|qq + (6 − q)|u|66 < 0,

which holds because λ, η > 0 and 4 < q < 6. Hence, u ∈ N− and the proof is complete. □

Remark 3.5. Since any critical point of I belongs to N−, if u is a critical point of I, it must be
a ground state solution to problem (1.1).



10 S. DENG, G. ZENG EJDE-2025/93

Lemma 3.6. Assume that a > 0. If 4 < q < 6, then there exists a u∗ ∈ X0(Ω) such that

sup
t≥0

I(tu∗) < c(A), (3.18)

where c(A) is the positive constant given in Lemma 3.3

Proof. Let U(x) = 31/4

(1+|x|2)1/2 . Then we know from [5] that U is a minimizer of Sn. Assume that

Bδ ⊂ Ω ⊂ B2δ and let ω ∈ C∞
0 (Ω) be such that 0 ≤ ω ≤ 1, ω(x) = 1 in Bδ and ω(x) = 0 in R3 \Ω.

For ϵ > 0 we define

Uϵ(x) := ϵ−1/2U(
x

ϵ
) and uϵ := ω(x)Uϵ(x).

Then from [18], we know that as ϵ→ 0,

∥uϵ∥2 = S3/2
n +O(ϵ), (3.19)

|uϵ|66 = S3/2
n +O(ϵ3), (3.20)

|uϵ|qq =


O(ϵq/2), 1 ≤ q < 3,

O(ϵq/2| ln ϵ|), q = 3,

O(ϵ3−
q
2 ), 3 < q < 6,

(3.21)

[uϵ] = O(ϵ2−2s), (3.22)

and ∫
Ω

|uϵ|q ln |uϵ|2 dx = Cϵ3−
q
2 ln

C

ϵ
+O(ϵ3−

q
2 ) +O(ϵq/2 ln ϵ). (3.23)

Now we consider the following two cases:

Case 1. s ≤ 1
2 Given the definitions of the functions I(u) and Ju(t), it is evident that limt→0 Juϵ

(t) =
0 and limt→∞ Juϵ

(t) = −∞. This behavior is consistent across all ϵ values in the interval
(0, ϵ0), where ϵ0 is a positive number that is sufficiently small but fixed. Thus, there exist
0 < t1 < t2 < +∞, independent of ϵ, such that

I(tuϵ) < c(A), ∀t ∈ (0, t1] ∪ [t2,+∞).

For t ∈ [t1, t2], we have

I(tuϵ) =
a

2
t2∥uϵ∥2X0

+
b

4
t4∥uϵ∥4 +

λ

4
t4
∫
Ω

ϕuϵ
u2ϵ dx+

2ηtq

q2
|uϵ|qq

− ηtq

q

∫
Ω

|uϵ|q ln |tuϵ|2 dx− t6

6
|uϵ|66

≤ a

2
t2∥uϵ∥2X0

+
b

4
t4∥uϵ∥4 +

λ

4
t4|ϕϵ|6|uϵ|212

5
+

2η

q2
tq|uϵ|qq

− η

q
tq
∫
Ω

|uϵ|q ln |tuϵ|2 dx− 1

6
t6|uϵ|66

≤ max
t∈[t1,t2]

g(t) +
λ

4
t4|ϕϵ|6|uϵ|212

5
+

2η

q2
tq|uϵ|qq −

η

q
tq
∫
Ω

|uϵ|q ln |tuϵ|2 dx

≤ max
t>0

g(t) + C1|ϕϵ|6|uϵ|212
5

+ C2|uϵ|qq − C3

∫
Ω

|uϵ|q ln |uϵ|2 dx,

(3.24)

where

g(t) :=
a

2
t2∥uϵ∥2X0

+
b

4
t4∥uϵ∥4 −

1

6
t6|uϵ|66.

Since limt→0 g(t) = 0 and limt→+∞ g(t) = −∞, so there exists a unique tϵ > 0 such that
maxt>0 g(t) = g(tϵ) and g′(tϵ) = tϵ(a∥uϵ∥2X0

+ bt2ϵ∥uϵ∥4 − t4ϵ |uϵ|66) = 0, where

t2ϵ =
b∥uϵ∥4 +

√
b2∥uϵ∥8 + 4a∥uϵ∥2X0

|uϵ|66
2|uϵ|66

.



EJDE-2025/93 SCHRÖDINGER-KIRCHHOFF-POISSON TYPE SYSTEMS 11

Using the estimates in (3.19)-(3.22), combining 1 < 2 − 2s, we have ∥uϵ∥X0
= ∥uϵ∥ as ϵ → 0.

Hence, one sees that as ϵ→ 0,

t2ϵ∥uϵ∥2X0
=
b∥uϵ∥6 +

√
b2∥uϵ∥12 + 4a∥uϵ∥6X0

|uϵ|66
2|uϵ|66

=
bS3

n +
√
b2S6

n + 4aS3
n

2
+O(ϵ)

= A+O(ϵ),

(3.25)

and

t6ϵ |uϵ|66 =
(b∥uϵ∥4 +

√
b2∥uϵ∥8 + 4a∥uϵ∥2X0

|uϵ|66
2|uϵ|46

)3

=
(bS2

n +
√
b2S4

n + 4aSn

2
+O(ϵ)

)3

=
A3

S3
n

+O(ϵ).

(3.26)

It follows from (3.25), (3.26) and the definition of g(t) that

max
t>0

g(t) = g(tϵ) = c(A) +O(ϵ), as ϵ→ 0. (3.27)

From (3.21), (3.23), (3.24) and (3.27), we have that

I(tuϵ) ≤ c(A) +O(ϵ) +O(ϵ) +O(ϵ3−
q
2 ) − Cϵ3−

q
2 ln

C

ϵ
+O(ϵq/2 ln ϵ), as ϵ→ 0, (3.28)

for t ∈ [t1, t2]. Since 4 < q < 6, we see that 3 − q
2 < 1 < q

2 , together with

lim
ϵ→0

ϵ3−
q
2

ϵ3−
q
2 ln C

ϵ

= 0

shows that

O(ϵ) +O(ϵ) +O(ϵ3−
q
2 ) − Cϵ3−

q
2 ln

C

ϵ
+O(ϵq/2 ln ϵ) < 0 (3.29)

for suitably small ϵ. Fix such an ϵ > 0. It then follows from (3.28) and (3.29) that

I(tuϵ) < c(A), ∀t ∈ [t1, t2].

Case 2. s > 1/2 Since limt→0 Juϵ
(t) = 0 and limt→∞ Juϵ

(t) = −∞, so there exists tη verifying
I(tηuϵ) = maxt≥0 I(tuϵ). Hence

at2η∥uϵ∥2X0
+ bt4η∥uϵ∥4 + λt4η

∫
Ω

ϕuϵu
2
ϵ dx− η

∫
Ω

tqη|uϵ|q ln |tηuϵ|2 dx− t6η|uϵ|66 = 0,

so we have

at2η∥uϵ∥2X0
+ bt4η∥uϵ∥4 + λt4η

∫
Ω

ϕuϵ
u2ϵ dx+

∣∣∣η ∫
Ω

tqη|uϵ|q ln |tηuϵ|2 dx
∣∣∣ ≥ t6η|uϵ|66,

which implies that tη is bounded. We claim that

tη → 0, as η → ∞. (3.30)

Arguing by contradiction, we can assume that there exists t0 > 0 and a sequence ηn with ηn → ∞
such that tηn

→ 0 as n→ ∞. Consequently, there is M > 0 such that

at2ηn
∥uϵ∥2X0

+ bt4ηn
∥uϵ∥4 + λt4ηn

∫
Ω

ϕuϵu
2
ϵ dx ≤M as n→ ∞,

for λ <∞ and so

ηn

∫
Ω

tqηn
|uϵ|q ln |tηnuϵ|2 dx− t6ηn

|uϵ|66 = ∞ as n→ ∞,



12 S. DENG, G. ZENG EJDE-2025/93

which is absurd. Indeed, by combining (3.24) with (3.30), we have

0 ≤ max
t≥0

I(tuϵ) = I(tηuϵ) → 0 as η → ∞,

and this readily implies the existence of η∗ > 0 such that

max
t≥0

I(tuϵ) < c(A), for all η > η∗

provided that ϵ > 0 is small enough but fixed. Taking u∗ = uϵ, we see that (3.18) is valid. The
proof is complete. □

Proof of Theorem 1.1. By combining Lemmas 3.2, 3.3, and 3.6, we have that there exists a u ∈
X0(Ω) such that u is a weak solution to problem (1.1). Furthermore, from Remark 3.5, we know
that the mountain pass type solution u is a ground state solution to problem 1.1. This proof is
complete. □

The proof of Corollary 1.2 is similar to Theorem 1.1, we omit it here.
The proof Corollary 1.3 is similar to Theorem 1.1, we omit it here.

Acknowledgements. This research was supported by National Natural Science Foundation of
China (No. 12371121).

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Shengbing Deng (corresponding author)

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China
Email address: shbdeng@swu.edu.cn

Guorong Zeng
School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

Email address: grzengmath@163.com


	1. Introduction and statement of main result
	2. Preliminaries and variational setting
	3. Proof of main result
	Acknowledgements

	References