Existence of Ground State Solutions to Nonlinear Schrödinger-Poisson System with Doping Profile and Coercive Potential

Xiaodong HAO; Shuibo HUANG; Huanhuan ZHANG

doi:10.1051/wujns/2026313281

Open Access

Issue		Wuhan Univ. J. Nat. Sci. Volume 31, Number 3, June 2026


Page(s)		281 - 290
DOI		https://doi.org/10.1051/wujns/2026313281
Published online		24 June 2026

Wuhan University Journal of Natural Sciences, 2026, Vol.31 No.3, 281-290

Mathematics

CLC number: O175.29

Existence of Ground State Solutions to Nonlinear Schrödinger-Poisson System with Doping Profile and Coercive Potential

含掺杂分布与强制势的非线性薛定谔-泊松系统基态解的存在性

Xiaodong HAO (郝晓冬), Shuibo HUANG (黄水波)^† and Huanhuan ZHANG (张环环)

School of Mathematics and Computer Science, Northwest Minzu University, Lanzhou 730030, Gansu, China (西北民族大学　数学与计算机科学学院，甘肃　兰州　730030)

^† Corresponding author. E-mail: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 31 December 2025

Abstract

This paper investigates the existence of ground state solutions to the following nonlinear Schrödinger-Poisson system by variational methods ${\begin{array}{l} - Δ u + ω u + V (x) u + e ϕ u = {| u |}^{p - 2} u, x \in R^{3}, \\ - Δ ϕ = \frac{e}{2} (u^{2} - ρ (x)), x \in R^{3}, \end{array}$ Mathematical equation provided that $ρ (x) \geq 0$ and $e^{2} {‖ ρ ‖}_{\frac{6}{5}} \leq ρ_{0}$ with $ρ_{0} > 0$ small enough, where $4 0$ denotes a coupling constant, $ω > 0$ is a Lagrange multiplier, the potential $V (x)$ is coercive and the doping profile $ρ (x)$ Mathematical equation satisfies appropriate decay conditions. The introduction of $ρ (x)$ compromises the coercivity of the energy functional. Therefore, we establish the existence of ground state solutions by considering the minimization problem on Nehari manifold.

摘要

本文利用变分方法研究了当满足条件 $ρ (x) \geq 0$ Mathematical equation 且 $e^{2} {‖ ρ ‖}_{\frac{6}{5}} \leq ρ_{0}$ ( $ρ_{0} > 0$ 为足够小的常数)时下述非线性薛定谔-泊松系统基态解的存在性： ${\begin{array}{l} - Δ u + ω u + V (x) u + e ϕ u = {| u |}^{p - 2} u, x \in R^{3}, \\ - Δ ϕ = \frac{e}{2} (u^{2} - ρ (x)), x \in R^{3}, \end{array}$ Mathematical equation 其中 $4 0$ 为耦合常数， $ω > 0$ 为拉格朗日乘子，势函数 $V (x)$ 具有强制性，掺杂分布 $ρ (x)$ 满足适当的衰减条件。 $ρ (x)$ 的引入破坏了能量泛函的强制性。因此，本文通过考虑奈哈里流形上的极小化问题来研究基态解的存在性。

Key words: doping profile / ground state solution / Nehari manifold / variational methods

关键字 : 掺杂分布 / 基态解 / 奈哈里流形 / 变分法

Cite this article: HAO Xiaodong, HUANG Shuibo, ZHANG Huanhuan. Existence of Ground State Solutions to Nonlinear Schrödinger-Poisson System with Doping Profile and Coercive Potential[J]. Wuhan Univ J of Nat Sci, 2026, 31(3): 281-290.

Biography: HAO Xiaodong, male, Master candidate, research direction: partial differential equations. E-mail: This email address is being protected from spambots. You need JavaScript enabled to view it.

Foundation item: Supported by the National Natural Science Foundation of China (12361026) and the Discipline Construction Fund Project of Northwest Minzu University

© Wuhan University 2026

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

0 Introduction

In this paper, we investigate the following nonlinear Schrödinger-Poisson system

${\begin{array}{l} - Δ u + ω u + V (x) u + e ϕ u = {| u |}^{p - 2} u, x \in R^{3}, \\ - Δ ϕ = \frac{e}{2} (u^{2} - ρ (x)), x \in R^{3}, \end{array}$ Mathematical equation (1)

where $4 0, ω > 0$ Mathematical equation appears as a Lagrange multiplier, $V (x)$ is the potential function satisfying assumptions:

$\begin{array}{l} (V_{1}) V (x) \in C^{1} (R^{3}, R), V (x) \geq 0 o n R^{3}, \\ (V_{2}) \underset{| x | \to \infty}{l i m} V (x) = \infty . \end{array}$ Mathematical equation

The doping profile $ρ (x)$ Mathematical equation satisfies assumptions:

$\begin{array}{l} (ρ_{1}) ρ (x) \in L^{\frac{6}{5}} (R^{3}), ρ (x) \geq 0, a n d x \cdot \nabla ρ (x) \in L^{\frac{6}{5}} (R^{3}), \\ (ρ_{2}) t h e r e e x i s t γ > 2 a n d C > 0 s u c h t h a t ρ (x) \leq \frac{C}{1 + | x |^{γ}} \end{array}$ Mathematical equation

for $x \in R^{3} .$ Mathematical equation

The presence of the potential $V (x)$ Mathematical equation significantly influences the functional-analytic framework. If $V (x)$ is non-radial, unbounded or exhibits complex decay, the compactness of Sobolev embeddings may be compromised, hindering the verification of critical points. Furthermore, if $V (x)$ tends to negative infinity or possesses deep potential wells, the energy functional may become unbounded below, necessitating additional constraints on $V (x)$ Mathematical equation to ensure well-posedness.

The nonlinear Schrödinger equation arises in quantum mechanics, nonlinear optics and other fields^[1]. As an important extension, the Schrödinger-Poisson system combines the interplay between quantum mechanics and electromagnetism, providing a more accurate description of charged particle motion in electromagnetic fields, see Refs. [2-4].

We first review relevant results for the case $V (x) = 0$ Mathematical equation and $ρ (x) = 0 .$ Consider the system

${\begin{array}{l} - Δ u + ω u + ϕ (x) u = u^{p}, \\ - Δ ϕ = u^{2}, \underset{| x | \to + \infty}{l i m} ϕ (x) = 0 . \end{array}$ Mathematical equation (2)

Ruiz^[5] proved that there exists a positive radial solution to problem (2) with Lagrange multiplier $ω = 1$ Mathematical equation provided $2 < p < 5$ . He further showed that no nontrivial solutions exist if $p \leq 2$ . Similar equations have been studied in Refs. [6-8].

When $V (x) \neq 0$ Mathematical equation and $ρ (x) = 0$ , the following nonlinear Schrödinger-Poisson system

${\begin{array}{l} - Δ u + V (x) u + λ ϕ (x) u = {| u |}^{p - 1} u, x \in R^{3}, \\ - Δ ϕ = u^{2}, \underset{| x | \to + \infty}{l i m} ϕ (x) = 0, x \in R^{3}, \end{array}$ Mathematical equation (3)

has been extensively studied. Specifically, for the coercive potential $V (x) (\underset{| x | \to \infty}{l i m} V (x) = \infty)$ Mathematical equation , Jiang et al^[9] demonstrated that problem (3) admits a ground state solution when $1 < p < 2$ provided that the coupling constant $λ$ is small enough. In the case where $V (x) = 1 + μ g (x)$ with

$g (x) \in L^{\infty} (R^{3})$ Mathematical equation , Jiang and Zhou^[10] established the existence of a ground state solution to problem (3) with $1 < p < 2$ . Furthermore, in Ref. [11], they studied the existence of positive solutions to problem (3) with

$V_{λ} (x) = λ + \frac{1}{| y |^{a}} (0 \leq a \leq 8) a n d m a x {2, \frac{2 + α}{2}} < p < 5 .$ Mathematical equation

Similarly, Azzollini and Pomponio^[12] investigated the following nonlinear Schrödinger-Maxwell equations

${\begin{array}{l} - Δ u + V (x) u + ϕ u = {| u |}^{p - 1} u, x \in R^{3}, \\ - Δ ϕ = u^{2}, x \in R^{3}, \end{array}$ Mathematical equation (4)

They proved that when $V (x)$ Mathematical equation is a positive constant, problem (4) admits a ground state solution provided $2 \leq p < 5$ . For a nonconstant potential $V (x)$ , they showed problem (4) admits a ground state solution under appropriate conditions on $V (x)$ provided $3 \leq p < 5$ . For other relevant studies on the existence of solutions to nonlinear Schrödinger equations involving potential functions, see Refs. [13-19].

When $V (x) = 0$ Mathematical equation and $ρ (x) \neq 0$ , Shan et al^[20] investigated the following nonlinear Schrödinger-Newton system

${\begin{array}{l} - Δ u + u = Φ u, x \in R^{3}, \\ - Δ Φ = u^{2} + ρ (x), x \in R^{3}, \end{array}$ Mathematical equation (5)

where $ρ \in L^{\frac{6}{5}} (R^{3})$ Mathematical equation is a doping profile. They demonstrated that problem (5) admits a positive ground state solution if $ρ (x) \geq 0$ and ${‖ ρ ‖}_{\frac{6}{5}}$ is small enough.

Recently, based on Ref. [7], Colin and Watanabe^[21] investigated the following nonlinear Schrödinger-Poisson system

${\begin{array}{l} - Δ u + ω u + e ϕ u = {| u |}^{p - 1} u, x \in R^{3}, \\ - Δ ϕ = \frac{e^{2}}{2} ({| u |}^{2} - ρ (x)), x \in R^{3}, \end{array}$ Mathematical equation (6)

where $ω > 0, e > 0$ Mathematical equation and $2 0$ (independent of $e$ and $ρ$ ) such that

$e^{2} ({‖ ρ ‖}_{\frac{6}{5}} + {‖ x \cdot \nabla ρ ‖}_{\frac{6}{5}} + {‖ x \cdot (D^{2} ρ x) ‖}_{\frac{6}{5}}) \leq ρ_{0} .$ Mathematical equation

Furthermore, Colin and Watanabe^[22] studied the existence of stable standing waves to problem (6) in the case of $1 < p < \frac{7}{3}$ Mathematical equation .

It is noteworthy that investigations involving both the doping profile $ρ (x)$ Mathematical equation and the potential $V (x)$ remain relatively scarce. Motivated by relevant studies^[9,12,21], the main purpose of this paper is to establish the existence of ground state solutions to problem (1). We employ variational methods, constructing an energy functional and analyzing its critical points via the Mountain Pass Lemma^[23] and the Palais-Smale condition^[24].

Define

$E : = {u \in H^{1} (R^{3}) : \int_{R^{3}} V (x) u^{2} d x < \infty},$ Mathematical equation

equipped with the inner product and norm

$\begin{array}{l} {〈 u, v 〉}_{E} = \int_{R^{3}} (\nabla u \nabla v + V (x) u v) d x, \\ {‖ u ‖}^{2} = \int_{R^{3}} ({| \nabla u |}^{2} + V (x) u^{2}) d x . \end{array}$ Mathematical equation (7)

The energy functional associated with (1) is given by

$\begin{array}{l} J (u) = \frac{1}{2} {\int_{R^{3}} | \nabla u |}^{2} d x + \frac{ω}{2} \int_{R^{3}} u^{2} d x + \frac{1}{2} \int_{R^{3}} V (x) u^{2} d x \\ - \frac{1}{p} {\int_{R^{3}} | u |}^{p} d x + e^{2} D (u), \end{array}$ Mathematical equation (8)

where $D (u)$ Mathematical equation is defined as

$\begin{array}{l} D (u) : = \frac{1}{4} \int_{R^{3}} F (u) (u^{2} - ρ (x)) d x \\ = \frac{1}{32 π} \int_{R^{3}} \int_{R^{3}} \frac{(u^{2} (x) - ρ (x)) (u^{2} (y) - ρ (y))}{| x - y |} d x d y, \end{array}$ Mathematical equation (9)

the nonlocal term $F (u)$ Mathematical equation is defined as

$F (u) (x) = F_{1} (u) (x) + F_{2} (x),$ Mathematical equation (10)

where

$F_{1} (u) (x) : = {(- Δ)}^{- 1} (\frac{{| u |}^{2}}{2}) = \frac{1}{8 π | x |} {| u |}^{2},$ Mathematical equation (11)

$F_{2} (x) : = {(- Δ)}^{- 1} (\frac{- ρ (x)}{2}) = - \frac{1}{8 π | x |} ρ (x) .$ Mathematical equation (12)

Definition 1 We say that $u$ Mathematical equation is a ground state solution to system (1) if $u$ has a least energy among all nontrivial solutions of (1), namely $u$ satisfies

$J (u) = i n f {J (u) | u \in H^{1} (R^{3}), J^{'} (u) = 0} .$ Mathematical equation (13)

The following is the main result of this paper.

Theorem 1 Assume that $4 < p < 6$ Mathematical equation , $V (x)$ satisfies $(V_{1}) - (V_{2})$ and $ρ$ satisfies $(ρ_{1}) - (ρ_{2})$ . Then system (1) admits a ground state solution provided that there exists a small enough constant $ρ_{0} > 0$ such that

$e^{2} {‖ ρ ‖}_{\frac{6}{5}} \leq ρ_{0}$ Mathematical equation (14)

Remark 1 The coexistence of $V (x)$ Mathematical equation and $ρ (x)$ complicates proving the Palais-Smale condition and Mountain Pass Lemma. Specifically, in verifying the Palais-Smale condition, although the coercivity of $V (x)$ is employed to overcome the lack of compactness on $R^{3}$ , the nonlocal terms introduced by $ρ (x)$ Mathematical equation may disrupt this compactness if ${‖ ρ ‖}_{\frac{6}{5}}$ is too large. Hence, the smallness condition $e^{2} {‖ ρ ‖}_{\frac{6}{5}} \leq ρ_{0}$ is crucial to the variational framework.

Remark 2 When $2 < p \leq 4$ Mathematical equation , there are contradictions in proving the Palais-Smale condition and the Mountain Pass Lemma. Therefore, we only consider the case where $4 < p < 6$ .

The paper is organized as follows: In Section 1, we present the necessary preliminary knowledge for subsequent proofs, including the decomposition of the energy functional, estimates of nonlocal terms, scaling properties of nonlocal terms and derivation of Nehari identity and Pohožaev identity. Section 2 is devoted to the proof of our main result. We first verify the boundedness of the functional and the Palais-Smale condition, then employ the Mountain Pass Lemma to prove the existence of a ground state solution (Theorem 1).

1 Preliminaries

The main purpose of this section is to present some lemmas and theorems required for the subsequent analysis, which closely follows the framework established in Refs. [21-22].

Firstly, rewrite the energy functional $J (u)$ Mathematical equation in the following way:

$\begin{array}{l} J (u) = \frac{1}{2} \int_{R^{3}} {| \nabla u |}^{2} d x + \frac{ω}{2} \int_{R^{3}} u^{2} d x + \frac{1}{2} \int_{R^{3}} V (x) u^{2} d x \\ - \frac{1}{p} \int_{R^{3}} {| u |}^{p} d x + \frac{e^{2}}{4} \int_{R^{3}} F_{1} (u) {| u |}^{2} d x + \frac{e^{2}}{4} \int_{R^{3}} F_{2} {| u |}^{2} d x \\ - \frac{e^{2}}{4} \int_{R^{3}} F_{1} (u) ρ (x) d x - \frac{e^{2}}{4} \int_{R^{3}} F_{2} ρ (x) d x \\ = \frac{1}{2} \int_{R^{3}} {| \nabla u |}^{2} d x + \frac{ω}{2} \int_{R^{3}} u^{2} d x + \frac{1}{2} \int_{R^{3}} V (x) u^{2} d x \\ - \frac{1}{p} \int_{R^{3}} {| u |}^{p} d x + e^{2} D_{1} (u) + 2 e^{2} D_{2} (u) + e^{2} D_{4}, \end{array}$ Mathematical equation

where $F_{1} (u), F_{2}$ Mathematical equation are defined by (11) and (12) respectively and

$D_{1} (u) = \frac{1}{4} \int_{R^{3}} F_{1} (u) {| u |}^{2} d x,$ Mathematical equation (15)

$D_{2} (u) = - \frac{1}{4} \int_{R^{3}} F_{1} (u) ρ (x) d x,$ Mathematical equation (16)

$D_{3} (u) = \frac{1}{4} \int_{R^{3}} F_{2} {| u |}^{2} d x,$ Mathematical equation (17)

$D_{4} = - \frac{1}{4} \int_{R^{3}} F_{2} ρ (x) d x,$ Mathematical equation (18)

$D_{5} (u) = \frac{1}{2} \int_{R^{3}} F_{1} (u) x \cdot \nabla ρ (x) d x,$ Mathematical equation (19)

$D_{6} = \frac{1}{2} \int_{R^{3}} F_{2} x \cdot \nabla ρ (x) d x .$ Mathematical equation (20)

Obviously, by (11) and (12),

$D_{2} (u) = D_{3} (u) .$ Mathematical equation (21)

The following lemma is devoted to present some estimates for the nonlocal terms in functional $J (u)$ Mathematical equation .

Lemma 1 For any $u \in E, F_{1}, F_{2}, D_{i}$ Mathematical equation (i=1, 2, 4, 5, 6) satisfy the following estimates:

${‖ F_{1} (u) ‖}_{6} \leq C {‖ u ‖}_{\frac{12}{5}}^{2} \leq C {‖ u ‖}^{2},$ Mathematical equation (22)

${‖ F_{2} ‖}_{6} \leq C {‖ ρ (x) ‖}_{\frac{6}{5}},$ Mathematical equation (23)

$D_{1} (u) \leq C {‖ u ‖}_{\frac{12}{5}}^{4} \leq C {‖ u ‖}^{4},$ Mathematical equation (24)

$| D_{2} (u) | \leq C {‖ u ‖}_{\frac{12}{5}}^{2} {‖ ρ (x) ‖}_{\frac{6}{5}} \leq C {‖ u ‖}^{2} {‖ ρ (x) ‖}_{\frac{6}{5}},$ Mathematical equation (25)

$| D_{4} | \leq C {‖ ρ (x) ‖}_{\frac{6}{5}}^{2},$ Mathematical equation (26)

$| D_{5} (u) | \leq C {‖ u ‖}_{\frac{12}{5}}^{2} {‖ x \cdot \nabla ρ (x) ‖}_{\frac{6}{5}} \leq C {‖ u ‖}^{2} {‖ x \cdot \nabla ρ (x) ‖}_{\frac{6}{5}},$ Mathematical equation (27)

$| D_{6} | \leq C {‖ ρ (x) ‖}_{\frac{6}{5}} ‖ x \cdot \nabla ρ (x) ‖_{\frac{6}{5}} .$ Mathematical equation (28)

Proof The proofs of (22), (24) and (25) can be found in Lemma 2.2 in Ref. [21]. The proofs of (23), (26), (27) and (28) can be found in Lemma 2.1 in Ref. [22].

We derive the scaling properties of the nonlocal terms $D_{1} (u)$ Mathematical equation and $D_{2} (u)$ , which is essential in the subsequent proof of the Pohožaev identity.

For $a, b \in R$ Mathematical equation and $λ > 0$ , denote $u_{λ} (x) : = λ^{a} u (λ^{b} x)$ .

Recall that

$F_{1} (u) (x) = {(- Δ)}^{- 1} (\frac{{| u (x) |}^{2}}{2}) = \frac{1}{8 π} \int_{R^{3}} \frac{{| u (y) |}^{2}}{| x - y |} d y .$ Mathematical equation

Let $z = λ^{b} y$ Mathematical equation , we have

$\begin{array}{l} F_{1} (u_{λ}) (x) = \frac{1}{8 π} \int_{R^{3}} \frac{{| u_{λ} (y) |}^{2}}{| x - y |} d y = \frac{λ^{2 a}}{8 π} \int_{R^{3}} \frac{{| u (λ^{b} y) |}^{2}}{| x - y |} d y \\ = \frac{λ^{2 a + b}}{8 π} \int_{R^{3}} \frac{{| u (λ^{b} y) |}^{2}}{| λ^{b} x - λ^{b} y |} d y \\ = \frac{λ^{2 a - 2 b}}{8 π} \int_{R^{3}} \frac{{| u (z) |}^{2}}{| λ^{b} x - z |} d z = λ^{2 a - 2 b} F_{1} (u) (λ^{b} x) . \end{array}$ Mathematical equation (29)

Thus

$\begin{array}{l} D_{1} (u_{λ}) = \frac{1}{4} \int_{R^{3}} F_{1} (u_{λ}) (x) {| u_{λ} (x) |}^{2} d x \\ = \frac{λ^{4 a - 2 b}}{4} \int_{R^{3}} F_{1} (u) (λ^{b} x) {| u (λ^{b} x) |}^{2} d x \\ = \frac{λ^{4 a - 5 b}}{4} \int_{R^{3}} F_{1} (u) (x) {| u (x) |}^{2} d x \\ = λ^{4 a - 5 b} D_{1} (u) . \end{array}$ Mathematical equation (30)

Similarly,

$\begin{array}{l} D_{2} (u_{λ}) = - \frac{1}{4} \int_{R^{3}} F_{1} (u_{λ}) ρ (x) d x \\ = - \frac{λ^{2 a - 2 b}}{4} \int_{R^{3}} F_{1} (u) (λ^{b} x) ρ (x) d x \\ = - \frac{λ^{2 a - 5 b}}{4} \int_{R^{3}} F_{1} (u) ρ (λ^{- b} x) d x . \end{array}$ Mathematical equation (31)

By the Hölder inequality and (22), it follows that

$\begin{array}{l} | D_{2} (u_{λ}) | \leq \frac{λ^{2 a - 5 b}}{4} {‖ F_{1} (u) ‖}_{6} {‖ ρ (λ^{- b} x) ‖}_{\frac{6}{5}} \\ \leq C λ^{2 a - \frac{5}{2} b} {‖ ρ ‖}_{\frac{6}{5}} {‖ u ‖}_{\frac{12}{5}}^{2} . \end{array}$ Mathematical equation (32)

In order to prove the existence of solutions, we establish the Nehari identity and the Pohožaev identity associated with (1).

Lemma 2^[21] Suppose $u \in E$ Mathematical equation is a solution to problem (1). Then $u$ satisfies the Nehari identity $N (u) = 0$ and the Pohožaev identity $P (u) = 0$ , where

and

$\begin{array}{l} P (u) = \frac{1}{2} {\int_{R^{3}} | \nabla u |}^{2} d x + \frac{3 ω}{2} \int_{R^{3}} {| u |}^{2} d x \\ + \frac{3}{2} \int_{R^{3}} V (x) {| u |}^{2} d x + \frac{1}{2} \int_{R^{3}} u^{2} x \cdot \nabla V (x) d x \\ + \frac{5 e^{2}}{4} \int_{R^{3}} F (u) ({| u |}^{2} - ρ (x)) d x \\ - \frac{e^{2}}{2} \int_{R^{3}} F (u) x \cdot \nabla ρ (x) d x - \frac{3}{p} {\int_{R^{3}} | u |}^{p} d x . \end{array}$ Mathematical equation (34)

Proof Firstly, observe that, for any $φ \in E$ Mathematical equation ,

$\begin{array}{l} J^{'} (u) φ = \int_{ℝ^{3}} \nabla u \cdot \nabla φ d x + ω \int_{R^{3}} u φ d x \\ + \int_{R^{3}} V (x) u φ d x - {\int_{R^{3}} | u |}^{p - 1} φ d x \\ + \frac{e^{2}}{4} \int_{R^{3}} F^{'} (u) φ ({| u |}^{2} - ρ (x)) d x \\ + \frac{e^{2}}{2} \int_{R^{3}} F (u) u φ d x, \end{array}$ Mathematical equation (35)

where $F (u)$ Mathematical equation is defined by (10).

For any $u \in E$ Mathematical equation , by the definition of $F$ , we have

$F^{'} (u) = F_{1}^{'} (u) .$ Mathematical equation (36)

It is easy to check that, for any $φ \in E$ Mathematical equation ,

$\begin{array}{l} {F_{1}}^{'} (u) φ = \frac{1}{8 π} \underset{t \to 0}{l i m} \int_{R^{3}} \frac{{| u (y) + t φ (y) |}^{2} - {| u (y) |}^{2}}{t | x - y |} d y \\ = \frac{1}{8 π} \underset{t \to 0}{l i m} \int_{R^{3}} \frac{2 u (y) φ (y) + t φ^{2} (y)}{| x - y |} d y \\ = \frac{1}{4 π} \int_{R^{3}} \frac{u (y) φ (y)}{| x - y |} d y . \end{array}$ Mathematical equation (37)

Therefore

$\begin{array}{l} F^{'} (u) u = F_{1}^{'} (u) u = \frac{1}{4 π} \int_{R^{3}} \frac{u {(y)}^{2}}{| x - y |} d y \\ = {(- Δ)}^{- 1} (u^{2}) = 2 F_{1} (u) . \end{array}$ Mathematical equation (38)

Thus, by (21) and (38), we obtain

$\begin{array}{l} \frac{e^{2}}{4} \int_{R^{3}} F^{'} (u) u (u^{2} - ρ (x)) d x + \frac{e^{2}}{2} \int_{R^{3}} F (u) u^{2} d x \\ = \frac{e^{2}}{2} \int_{R^{3}} F_{1} (u) (u^{2} - ρ (x)) d x + \frac{e^{2}}{2} \int_{R^{3}} F_{1} (u) u^{2} d x \\ + \frac{e^{2}}{2} \int_{R^{3}} F_{2} {| u |}^{2} d x \\ = e^{2} \int_{R^{3}} F_{1} (u) u^{2} d x - \frac{e^{2}}{2} \int_{R^{3}} F_{1} (u) ρ (x) d x + \frac{e^{2}}{2} \int_{R^{3}} F_{2} u^{2} d x \\ = e^{2} \int_{R^{3}} F_{1} (u) u^{2} d x - e^{2} \int_{R^{3}} F_{1} (u) ρ (x) d x \\ = 4 e^{2} D_{1} (u) + 4 e^{2} D_{2} (u), \end{array}$ Mathematical equation

which leads to (33) holds.

Now, prove (34) by argument similarly as in Refs. [21,25].

Consider $u_{λ} (x) = u (\frac{x}{λ})$ Mathematical equation with $a = 0$ and $b = - 1$ . Then, from (30) and (31), we deduce that

$\begin{array}{l} J (u_{λ}) = \frac{1}{2} {\int_{R^{3}} | \nabla u_{λ} |}^{2} d x + \frac{ω}{2} \int_{R^{3}} u_{λ}^{2} d x + \frac{1}{2} \int_{R^{3}} V (x) u_{λ}^{2} d x \\ - \frac{1}{p} \int_{R^{3}} u_{λ}^{p} d x + e^{2} D_{1} (u_{λ}) + 2 e^{2} D_{2} (u_{λ}) + e^{2} D_{4} \\ = \frac{λ}{2} {\int_{R^{3}} | \nabla u |}^{2} d x + \frac{λ^{3} ω}{2} \int_{R^{3}} u^{2} d x + \frac{λ^{3}}{2} \int_{R^{3}} V (λ x) u^{2} d x \\ - \frac{λ^{3}}{p} \int_{R^{3}} u^{p} d x + e^{2} λ^{5} D_{1} (u) \\ - \frac{λ^{5} e^{2}}{2} \int_{R^{3}} F_{1} (u) ρ (λ x) d x + e^{2} D_{4}, \end{array}$ Mathematical equation (39)

where $D_{1} (u), D_{2} (u), D_{4}$ Mathematical equation are defined by (15), (16) and (18) respectively.

By $D (u) = D_{1} (u) + 2 D_{2} (u) + D_{4}$ Mathematical equation and (39), it follows that

$\begin{array}{l} 0 = \frac{d}{d λ} {J (u_{λ}) |}_{λ = 1} = \frac{1}{2} {\int_{R^{3}} | \nabla u |}^{2} d x + \frac{3 ω}{2} \int_{R^{3}} u^{2} d x \\ + \frac{3}{2} \int_{R^{3}} V (x) u^{2} d x + \frac{1}{2} \int_{R^{3}} u^{2} x \cdot \nabla V (x) d x \\ - \frac{3}{p} \int_{R^{3}} u^{p} d x + 5 e^{2} D_{1} (u) + 10 e^{2} D_{2} (u) - e^{2} D_{5} (u) \\ = \frac{1}{2} {\int_{R^{3}} | \nabla u |}^{2} d x + \frac{3 ω}{2} \int_{R^{3}} u^{2} d x \\ + \frac{3}{2} \int_{R^{3}} V (x) {| u |}^{2} d x + \frac{1}{2} \int_{R^{3}} u^{2} x \cdot \nabla V (x) d x \\ - \frac{3}{p} {\int_{R^{3}} | u |}^{p} d x - \frac{e^{2}}{2} \int_{R^{3}} F (u) x \cdot \nabla ρ (x) d x \\ + 5 e^{2} D (u) - 5 e^{2} D_{4} + e^{2} D_{6}, \end{array}$ Mathematical equation (40)

where $D_{5} (u), D_{6}$ Mathematical equation are defined by (19) and (20) respectively, which we use the fact that

$\frac{e^{2}}{2} \int_{R^{3}} F (u) x \cdot \nabla ρ (x) d x = e^{2} D_{5} (u) + e^{2} D_{6} .$ Mathematical equation

Denote

$\begin{array}{l} K = - 5 e^{2} D_{4} + e^{2} D_{6} \\ = \frac{5 e^{2}}{4} \int_{R^{3}} F_{2} (x) ρ (x) d x + \frac{e^{2}}{2} \int_{R^{3}} F_{2} (x) x \cdot \nabla ρ (x) d x . \end{array}$ Mathematical equation

By the definition of $F_{2}$ Mathematical equation (see (12)) and integration by parts, we have

$- Δ F_{2} (x) = \frac{- ρ (x)}{2} a n d \int_{R^{3}} {| \nabla F_{2} |}^{2} d x = - \frac{1}{2} \int_{R^{3}} F_{2} ρ (x) d x .$ Mathematical equation

Thus

$\begin{array}{l} \int_{R^{3}} F_{2} x \cdot \nabla ρ d x \\ = - \int_{ℝ^{3}} \nabla F_{2} \cdot x ρ d x - 3 \int_{R^{3}} F_{2} ρ d x \\ = - 2 \int_{ℝ^{3}} \nabla F_{2} \cdot x Δ F_{2} d x - 3 \int_{R^{3}} F_{2} ρ d x \\ = 2 \int_{ℝ^{3}} \nabla (\nabla F_{2} \cdot x) \cdot \nabla F_{2} d x - 3 \int_{R^{3}} F_{2} ρ d x \\ = 2 \int_{R^{3}} {| \nabla F_{2} |}^{2} d x + 2 \int_{R^{3}} x \cdot \nabla (\frac{1}{2} | \nabla F_{2} |^{2}) d x - 3 \int_{R^{3}} F_{2} ρ d x \\ = 2 \int_{R^{3}} {| \nabla F_{2} |}^{2} d x - 3 \int_{R^{3}} {| \nabla F_{2} |}^{2} d x - 3 \int_{R^{3}} F_{2} ρ d x \\ = - \int_{R^{3}} {| \nabla F_{2} |}^{2} d x - 3 \int_{R^{3}} F_{2} ρ d x \\ = - \frac{5}{2} \int_{R^{3}} F_{2} ρ d x . \end{array}$ Mathematical equation

We deduce that $K = 0$ Mathematical equation . Using (40), we obtain (34).

Theorem 2^[26] Let $X$ Mathematical equation be a real Banach space with its dual space $X^{'}$ and suppose that $J \in C^{1} (X, R)$ satisfies

$m a x {J (0), J (ψ)} \leq ξ < η \leq \underset{‖ u ‖_{X} = ϱ}{i n f} J (u),$ Mathematical equation (41)

for some $ξ < η, ϱ > 0$ Mathematical equation and $ψ \in X$ with ${‖ ψ ‖}_{X} > ϱ$ . Let $c \geq η$ be characterized by

$c = \underset{γ \in Γ}{i n f} m a x_{s \in [0,1]} J (γ (s)),$ Mathematical equation (42)

where $Γ = {γ \in C ([0,1], X) : γ (0) = 0, γ (1) = ψ}$ Mathematical equation is the set of continuous paths joining 0 and $ψ$ . Then there exists a sequence ${u_{n}} \subset X$ such that

$J (u_{n}) \to c \geq η a n d (1 + {‖ u_{n} ‖}_{X}) {‖ J^{'} (u_{n}) ‖}_{X^{'}} \to 0 a s n \to \infty .$ Mathematical equation (43)

2 Proof of Theorem 1

In this section, we investigate the existence of ground state solutions when $4 < p < 6$ Mathematical equation and provide the proof of Theorem 1.

Definition 2 Let $E$ Mathematical equation be a real Banach space and $J : E \to R$ be a Fréchet differentiable functional, and its Fréchet derivative denoted by $J^{'} : E \to E^{*}$ (where $E^{*}$ is the dual space of $E$ ). The Nehari manifold associated with the energy functional $J$ is defined as follows

$N = {u \in E \ {0} | J^{'} (u) u = 0} .$ Mathematical equation

In order to establish our result, we adopt the following approach: we seek the minimizer of functional (8) constrained to the Nehari manifold

$N = {u \in E \ {0} | N (u) = 0},$ Mathematical equation

where

$\begin{array}{l} N (u) = {\int_{R^{3}} | \nabla u |}^{2} d x + ω \int_{R^{3}} u^{2} d x + \int_{R^{3}} V (x) u^{2} d x \\ - {\int_{R^{3}} | u |}^{p} d x + 4 e^{2} D_{1} (u) + 4 e^{2} D_{2} (u), \\ = {‖ u ‖}^{2} + ω {‖ u ‖}_{2}^{2} - {‖ u ‖}_{p}^{p} + 4 e^{2} D_{1} (u) + 4 e^{2} D_{2} (u) . \end{array}$ Mathematical equation (44)

Lemma 3^[12] Assume that $4 < p < 6,$ Mathematical equation $V (x)$ satisfies $(V_{1}) - (V_{2})$ and $ρ$ satisfies $(ρ_{1}) - (ρ_{2})$ . Then the following hold:

ⅰ) For any $u \neq 0$ Mathematical equation , there exists a unique number $\bar{t} > 0$ such that $\bar{t} u \in N$ and $J (\bar{t} u) = m a x_{t \geq 0} J (t u),$

ⅱ) there exists a positive constant $C$ Mathematical equation , such that for all $u \in N, {‖ u ‖}_{p} \geq C,$

ⅲ) $N$ Mathematical equation is a $C^{1}$ manifold,

provided that (14) holds with $ρ_{0}$ Mathematical equation small enough.

Proof Assertions ⅰ) and ⅱ) follow from standard arguments (see Ref. [27]). We now prove ⅲ).

For any $u \in E$ Mathematical equation , observe that

$\begin{array}{l} N (u) = 4 J (u) - {‖ u ‖}^{2} - ω {‖ u ‖}_{2}^{2} \\ - \frac{p - 4}{p} {‖ u ‖}_{p}^{p} - 4 e^{2} D_{2} (u) - 4 e^{2} D_{4} . \end{array}$ Mathematical equation

By (14), (25) and assertion ⅱ), for any $u \in N$ Mathematical equation , we have

$\begin{array}{l} N^{'} (u) u = - 2 {‖ u ‖}^{2} - 2 ω {‖ u ‖}_{2}^{2} - (p - 4) {‖ u ‖}_{p}^{p} - 4 e^{2} D_{2}^{'} (u) u \\ = - 2 {‖ u ‖}^{2} - 2 ω {‖ u ‖}_{2}^{2} - (p - 4) {‖ u ‖}_{p}^{p} - 8 e^{2} D_{2} (u) \\ \leq - 2 {‖ u ‖}^{2} + C e^{2} {‖ ρ ‖}_{\frac{6}{5}} {‖ u ‖}^{2} - C \\ \leq (- 2 + C ρ_{0}) {‖ u ‖}^{2} - C, \end{array}$ Mathematical equation

which we use the fact that $D_{2}^{'} (u) u = 2 D_{2} (u) .$ Mathematical equation

Choose $ρ_{0} > 0$ Mathematical equation small enough such that

$- 2 + C ρ_{0} < 0 .$ Mathematical equation

Thus, it follows that

$N^{'} (u) u \leq - C < 0 .$ Mathematical equation

The Nehari manifold $N$ Mathematical equation is a natural constraint for the functional $J$ . Therefore, we only need to seek critical points of $J$ on $N$ .

Now, denote the ground state energy of system (1) by

$c_{0} : = \underset{u \in N}{i n f} J (u)$ Mathematical equation

Lemma 4 Assume that $4 0$ such that for any $u \in N$ ,

$J (u) ⩾ δ_{1} {‖ u ‖}^{2}$ Mathematical equation (45)

provided that (14) holds with $ρ_{0} > 0$ Mathematical equation small enough.

Proof Since $u \in N$ Mathematical equation , observe that

${‖ u ‖}_{p}^{p} = {‖ u ‖}^{2} + ω {‖ u ‖}_{2}^{2} + 4 e^{2} D_{1} (u) + 4 e^{2} D_{2} (u) .$ Mathematical equation (46)

Combining with (46) and using (14) and (25), we obtain

$\begin{array}{l} J (u) = \frac{1}{2} {‖ u ‖}^{2} + \frac{ω}{2} {‖ u ‖}_{2}^{2} - \frac{1}{p} [{‖ u ‖}^{2} + ω {‖ u ‖}_{2}^{2} + 4 e^{2} D_{1} (u) + 4 e^{2} D_{2} (u)] + e^{2} D_{1} (u) \\ + 2 e^{2} D_{2} (u) + e^{2} D_{4} \\ = (\frac{1}{2} - \frac{1}{p}) ({‖ u ‖}^{2} + ω {‖ u ‖}_{2}^{2}) + (1 - \frac{4}{p}) e^{2} D_{1} (u) \\ + (2 - \frac{4}{p}) e^{2} D_{2} (u) + e^{2} D_{4} \\ \geq (\frac{1}{2} - \frac{1}{p}) {‖ u ‖}^{2} - C (2 - \frac{4}{p}) e^{2} {‖ ρ ‖}_{\frac{6}{5}} {‖ u ‖}^{2} \\ \geq [(\frac{1}{2} - \frac{1}{p}) - C (2 - \frac{4}{p}) ρ_{0}] {‖ u ‖}^{2} . \end{array}$ Mathematical equation (47)

Now, choose $ρ_{0} > 0$ Mathematical equation small enough such that

$(\frac{1}{2} - \frac{1}{p}) - C (2 - \frac{4}{p}) ρ_{0} > \frac{1}{4} (\frac{1}{2} - \frac{1}{p}) > 0 .$ Mathematical equation (48)

Substituting (48) into (47) yields

$J (u) \geq \frac{1}{4} (\frac{1}{2} - \frac{1}{p}) {‖ u ‖}^{2} \geq δ_{1} {‖ u ‖}^{2},$ Mathematical equation (49)

which completes the proof of (45).

Lemma 5 Assume that $4 0$ such that

$\underset{‖ u ‖ = σ}{i n f} J (u) > 0,$ Mathematical equation (50)

provided that (14) holds with $ρ_{0} > 0$ Mathematical equation small enough.

Proof It follows from Ref. [9] that the embedding $E \to H^{1} (R^{3})$ Mathematical equation is continuous. According to the embedding result in Ref. [15], we derive that

${‖ u ‖}_{q} \leq C ‖ u ‖,$ Mathematical equation (51)

where $2 \leq q < 6$ Mathematical equation .

From (14), (25) and (51), it follows that

$\begin{array}{l} J (u) |_{‖ u ‖ = σ} \geq \frac{1}{2} σ^{2} - \frac{C}{2} e^{2} {‖ ρ ‖}_{\frac{6}{5}} σ^{2} - \frac{C}{p} σ^{p} \\ \geq \frac{1}{2} σ^{2} - \frac{C}{2} ρ_{0} σ^{2} - \frac{C}{p} σ^{p} \\ = \frac{1}{8} σ^{2} + (\frac{3}{8} - \frac{C}{2} ρ_{0}) σ^{2} - \frac{C}{p} σ^{p} . \end{array}$ Mathematical equation (52)

We now choose the constant $ρ_{0} > 0$ Mathematical equation small enough such that

$\frac{3}{8} - \frac{C}{2} ρ_{0} > \frac{1}{4} .$ Mathematical equation

By (52), we get

$\begin{array}{l} J (u) |_{‖ u ‖ = σ} \geq \frac{1}{8} σ^{2} + \frac{1}{4} σ^{2} - \frac{C}{p} σ^{p} \\ = \frac{1}{8} σ^{2} + (\frac{1}{4} - \frac{C}{p} σ^{p - 2}) σ^{2} . \end{array}$ Mathematical equation (53)

Fix $ρ_{0}$ Mathematical equation , choose $σ > 0$ small enough such that

$\frac{1}{4} - \frac{C}{p} σ^{p - 2} > 0 .$ Mathematical equation

Consequently

$J (u) |_{‖ u ‖ = σ} \geq \frac{1}{8} σ^{2} + (\frac{1}{4} - \frac{C}{p} σ^{p - 2}) σ \geq \frac{1}{8} σ^{2} > 0,$ Mathematical equation (54)

which completes the proof of (50).

Lemma 6 (Energy tends to negative infinity) Assume that $4 < p < 6$ Mathematical equation . $V (x)$ satisfies $(V_{1}) - (V_{2})$ and $ρ$ satisfies $(ρ_{1}) - (ρ_{2})$ . Then there exists $u_{0} \in E \ {0}$ , such that

$J (t u_{0}) \to - \infty a s t \to \infty .$ Mathematical equation

Proof Fixing $u_{0} \in E \ {0}$ Mathematical equation , then

$\begin{array}{l} J (t u_{0}) = \frac{t^{2}}{2} {‖ u_{0} ‖}^{2} + \frac{ω t^{2}}{2} {‖ u_{0} ‖}_{2}^{2} - \frac{t^{p}}{p} {‖ u_{0} ‖}_{p}^{p} \\ + \frac{e^{2} t^{4}}{4} \int_{R^{3}} F_{1} (u_{0}) u_{0}^{2} d x - \frac{e^{2} t^{2}}{2} \int_{R^{3}} F_{1} (u_{0}) ρ (x) d x \\ + \frac{e^{2}}{4} \int_{R^{3}} F_{2} ρ (x) d x \\ \to - \infty, \end{array}$ Mathematical equation

as $t \to \infty$ Mathematical equation , since $4 < p < 6$ .

Theorem 3 Assume that $4 < p < 6$ Mathematical equation , $V (x)$ satisfies $(V_{1}) - (V_{2})$ and $ρ$ satisfies $(ρ_{1}) - (ρ_{2})$ , then the energy functional $J$ satisfies the Palais-Smale (PS) condition in $E$ ; that is, any sequence ${u_{n}} \subset E$ satisfying

$J (u_{n}) \to C a n d J^{'} (u_{n}) \to 0,$ Mathematical equation (55)

admits a strongly convergent subsequence provided that (14) holds with $ρ_{0} > 0$ Mathematical equation small enough.

Proof The proof is divided into four steps.

Step 1 Firstly, we prove that $u_{n}$ Mathematical equation is bounded in $E$ .

Let ${u_{n}} \subset E$ Mathematical equation satisfy

$J (u_{n}) \to C a n d J^{'} (u_{n}) \to 0 .$ Mathematical equation (56)

By (25), (33) and (14), we have

$\begin{array}{l} J (u_{n}) - \frac{1}{4} J^{'} (u_{n}) u_{n} \\ = \frac{1}{4} {‖ u_{n} ‖}^{2} + \frac{ω}{4} {‖ u_{n} ‖}_{2}^{2} + (\frac{1}{4} - \frac{1}{p}) {‖ u_{n} ‖}_{p}^{p} \\ + e^{2} D_{2} (u_{n}) + e^{2} D_{4} \\ \geq \frac{1}{4} {‖ u_{n} ‖}^{2} - C e^{2} {‖ ρ ‖}_{\frac{6}{5}} {‖ u_{n} ‖}^{2} \\ \geq (\frac{1}{4} - C ρ_{0}) {‖ u_{n} ‖}^{2} . \end{array}$ Mathematical equation

Choose $ρ_{0} > 0$ Mathematical equation small enough such that

$\frac{1}{4} - C ρ_{0} \geq \frac{1}{8} .$ Mathematical equation

Consequently

$\frac{1}{8} {‖ u_{n} ‖}^{2} \leq J (u_{n}) - \frac{1}{4} J^{'} (u_{n}) u_{n} \leq C .$ Mathematical equation

Therefore, the sequence ${u_{n}}$ Mathematical equation is bounded in $E$ . By the embedding $E \to H^{1} (R^{3})$ is continuous, then the sequence is also bounded in $H^{1} (R^{3})$ .

Step 2 Compactness analysis. Since ${u_{n}}$ Mathematical equation is bounded in $E$ and the embedding $E \to L^{q} (R^{3})$ is compact for $2 \leq q < 6$ . Therefore, there exists a subsequence (still denoted by ${u_{n}}$ ) such that

$u_{n} ⇀ u i n E,$ Mathematical equation

and

$u_{n} \to u i n L^{q} (R^{3}) .$ Mathematical equation (57)

Step 3 Non-local term convergence. We now establish the convergence of the nonlocal terms $D_{1} (u_{n})$ Mathematical equation and $D_{2} (u_{n})$ . The argument for $D_{2} (u_{n})$ is analogous to that for $D_{1} (u_{n})$ . Thus, we give the detailed proof for $D_{1} (u_{n})$ only.

By (22), the mapping $F_{1} : L^{\frac{12}{5}} (R^{3}) \to L^{6} (R^{3})$ Mathematical equation is continuous.

Applying (57) together with the Hölder inequality yields

$\begin{array}{l} {‖ F_{1} (u_{n}) - F_{1} (u) ‖}_{6} \leq C {‖ u_{n}^{2} - u^{2} ‖}_{\frac{6}{5}} \\ \leq C {‖ u_{n} - u ‖}_{\frac{12}{5}} ({‖ u_{n} ‖}_{\frac{12}{5}} + {‖ u ‖}_{\frac{12}{5}}) \to 0 . \end{array}$ Mathematical equation (58)

Combining (58) and the Hölder inequality, we deduce

$\begin{array}{l} | \int_{R^{3}} F_{1} (u_{n}) u_{n}^{2} - F_{1} (u) u^{2} d x | \\ \leq {‖ F_{1} (u_{n}) - F_{1} (u) ‖}_{6} {‖ u_{n}^{2} ‖}_{\frac{6}{5}} + {‖ F_{1} (u) ‖}_{6} {‖ u_{n}^{2} - u^{2} ‖}_{\frac{6}{5}} \\ \leq {‖ F_{1} (u_{n}) - F_{1} (u) ‖}_{6} {‖ u_{n}^{2} ‖}_{\frac{6}{5}} \\ + {‖ F_{1} (u) ‖}_{6} {‖ u_{n} - u ‖}_{\frac{12}{5}} ({‖ u_{n} ‖}_{\frac{12}{5}} + {‖ u ‖}_{\frac{12}{5}}) \\ \to 0 . \end{array}$ Mathematical equation

Recalling the definition of $D_{1} (u)$ Mathematical equation (see (15)), it follows that

$D_{1} (u_{n}) \to D_{1} (u) .$ Mathematical equation (59)

By a similar argument, we have

$D_{2} (u_{n}) \to D_{2} (u) .$ Mathematical equation (60)

Step 4 Strong convergence verification. We now prove that $u_{n} \to u$ Mathematical equation in $E$ . From the Palais-Smale condition $J^{'} (u_{n}) \to 0$ and $u_{n} ⇀ u$ in $E$ , we have

$\begin{array}{l} J^{'} (u_{n}) (u_{n} - u) \\ = \int_{ℝ^{3}} \nabla u_{n} \nabla (u_{n} - u) d x + \int_{R^{3}} V (x) u_{n} (u_{n} - u) d x \\ + ω \int_{R^{3}} u_{n} (u_{n} - u) d x - \int_{R^{3}} {| u_{n} |}^{p - 2} u_{n} (u_{n} - u) d x \\ + e^{2} \int_{R^{3}} F_{1} (u_{n}) u_{n} (u_{n} - u) d x + e^{2} \int_{R^{3}} F_{2} u_{n} (u_{n} - u) d x \\ = G_{1} + G_{2} + G_{3} - G_{4} + G_{5} + G_{6} \end{array}$ Mathematical equation

$= o (1) .$ Mathematical equation (61)

We proceed to analyze each term in the expansion of $J^{'} (u_{n}) (u_{n} - u)$ Mathematical equation separately.

$\begin{array}{l} G_{1} + G_{2} \\ = \int_{ℝ^{3}} \nabla u_{n} \nabla (u_{n} - u) d x + \int_{R^{3}} V (x) u_{n} (u_{n} - u) d x \\ = \int_{R^{3}} ({| \nabla u_{n} |}^{2} + V (x) {| u_{n} |}^{2}) d x - \int_{R^{3}} (\nabla u_{n} \nabla u + V (x) u_{n} u) d x \end{array}$ Mathematical equation

$= {‖ u_{n} ‖}^{2} - {〈 u_{n}, u 〉}_{E},$ Mathematical equation (62)

where ${〈 \cdot, \cdot 〉}_{E}$ Mathematical equation denotes the inner product in $E$ .

$G_{3} = ω \int_{R^{3}} u_{n} (u_{n} - u) d x = ω ({‖ u_{n} ‖}_{2}^{2} - \int_{R^{3}} u_{n} u d x) .$ Mathematical equation (63)

By (57) and Hölder inequality, we have

$\begin{array}{l} | G_{4} | = | {\int_{R^{3}} | u_{n} |}^{p - 2} u_{n} (u_{n} - u) d x | \leq {‖ {| u_{n} |}^{p - 1} ‖}_{\frac{p}{p - 1}} {‖ u_{n} - u ‖}_{p} \\ = {‖ u_{n} ‖}_{p}^{p - 1} {‖ u_{n} - u ‖}_{p} \end{array}$ Mathematical equation

$\leq C {‖ u_{n} - u ‖}_{p} \to 0 .$ Mathematical equation (64)

Similarly, applying the Hölder inequality together with (57) and (59), we obtain

$\begin{array}{l} | G_{5} | = | e^{2} \int_{R^{3}} F_{1} (u_{n}) u_{n} (u_{n} - u) d x | \\ \leq e^{2} {‖ F_{1} (u_{n}) ‖}_{6} {‖ u_{n} ‖}_{\frac{12}{5}} {‖ u_{n} - u ‖}_{\frac{12}{5}} \\ \leq C {‖ u_{n} - u ‖}_{\frac{12}{5}} \to 0 . \end{array}$ Mathematical equation (65)

An analogous argument, by (57), (60) and Hölder inequality, yields

$\begin{array}{l} | G_{6} | = | e^{2} \int_{R^{3}} F_{2} u_{n} (u_{n} - u) d x | \\ \leq C {‖ F_{2} ‖}_{6} {‖ u_{n} ‖}_{\frac{12}{5}} {‖ u_{n} - u ‖}_{\frac{12}{5}} \to 0 . \end{array}$ Mathematical equation (66)

Substituting the above results into (61), we get

$\begin{array}{l} J^{'} (u_{n}) (u_{n} - u) \\ = {‖ u_{n} ‖}^{2} - {〈 u_{n}, u 〉}_{E} + ω ({‖ u_{n} ‖}_{2}^{2} - \int_{R^{3}} u_{n} u d x) - G_{3} + G_{4} + G_{5} \\ = {‖ u_{n} ‖}^{2} - {〈 u_{n}, u 〉}_{E} + ω ({‖ u_{n} ‖}_{2}^{2} - 〈 u_{n}, u 〉) + o (1) = o (1) . \end{array}$ Mathematical equation (67)

Since $u_{n} ⇀ u$ Mathematical equation in $E$ and $u_{n} \to u$ in $L^{q} (R^{3})$ for $2 \leq q < 6$ , it follows that

${〈 u_{n}, u 〉}_{E} \to {‖ u ‖}^{2} a n d 〈 u_{n}, u 〉 \to {‖ u ‖}_{2}^{2} .$ Mathematical equation (68)

Substituting (68) into (67) yields

${‖ u_{n} ‖}^{2} - {‖ u ‖}^{2} + ω ({‖ u ‖}_{2}^{2} - {‖ u ‖}_{2}^{2}) = o (1) (n \to \infty) .$ Mathematical equation (69)

By the weak lower semicontinuity of norms

${‖ u ‖}^{2} \leq \underset{n \to \infty}{l i m i n f} {‖ u_{n} ‖}^{2} a n d {‖ u ‖}_{2}^{2} \leq \underset{n \to \infty}{l i m i n f} {‖ u_{n} ‖}_{2}^{2} .$ Mathematical equation (70)

From (69) and (70), it follows that the following conditions hold

$\underset{n \to \infty}{l i m} {‖ u_{n} ‖}^{2} = {‖ u ‖}^{2} a n d \underset{n \to \infty}{l i m} {‖ u_{n} ‖}_{2}^{2} = {‖ u ‖}_{2}^{2} .$ Mathematical equation

Consequently

$\begin{array}{l} {‖ u_{n} - u ‖}^{2} = {‖ u_{n} ‖}^{2} - 2 {〈 u_{n}, u 〉}_{E} + {‖ u ‖}^{2} \\ \to {‖ u ‖}^{2} - 2 {‖ u ‖}^{2} + {‖ u ‖}^{2} = 0 (n \to \infty) . \end{array}$ Mathematical equation (71)

Therefore, $u_{n} \to u$ Mathematical equation in $E$ , combined with the continuity of the embedding $E \to H^{1} (R^{3})$ , which implies $u_{n} \to u$ in $H^{1} (R^{3})$ .

Theorem 4 Assume that $4 < p < 6$ Mathematical equation . $V (x)$ satisfies $(V_{1}) - (V_{2})$ and $ρ$ satisfies $(ρ_{1}) - (ρ_{2})$ . Then system (1) admits aground state solution provided that there exists a small enough constant $ρ_{0} > 0$ such that (14) holds.

Proof By Lemma 5, there exist constants $ρ_{0}$ Mathematical equation small enough and $σ > 0$ , such that

$\underset{‖ u ‖ = σ}{i n f} J (u) > 0 .$ Mathematical equation

By Lemma 6, there exists $u_{0} \in E \ {0}$ Mathematical equation , such that

$J (t u_{0}) \to - \infty a s t \to \infty .$ Mathematical equation

Define $ψ = t_{0} u_{0}$ Mathematical equation for large enough $t_{0} > 0$ , we get

$J (ψ) < 0$ Mathematical equation

Thus, $J$ Mathematical equation satisfies the mountain pass geometric structure.

Define the mountain-pass level

$c = \underset{γ \in Γ}{i n f} m a x_{s \in [0,1]} J (γ (s)),$ Mathematical equation (72)

where $Γ = {γ \in C ([0,1], E) : γ (0) = 0, γ (1) = ψ} .$ Mathematical equation

By the Mountain Pass Lemma (Theorem 2), there exists a sequence ${u_{n}} \subset E$ Mathematical equation such that

$J (u_{n}) \to c \geq η a n d (1 + ‖ u_{n} ‖) {‖ J^{'} (u_{n}) ‖}_{E^{'}} \to 0 .$ Mathematical equation

Since $J$ Mathematical equation satisfies Palais-Smale condition, there exists $u \in E$ such that

$u_{n} \to u i n E,$ Mathematical equation

By the continuity of $J$ Mathematical equation and $J^{'}$ , we have

$J (u_{n}) \to J (u) = c \geq η > 0 a n d J^{'} (u_{n}) \to J^{'} (u) = 0 (n \to \infty) .$ Mathematical equation

Consequently, $u$ Mathematical equation is a critical point of $J$ .

We now prove that $u \neq 0$ Mathematical equation .

By contradiction, suppose that $u = 0$ Mathematical equation , it follow that

$J (u) = J (0) = e^{2} D_{4} .$ Mathematical equation

By (26) and (14), we get

$J (0) = e^{2} D_{4} \leq C {‖ ρ ‖}_{\frac{6}{5}}^{2} \leq C ρ_{0}^{2} .$ Mathematical equation

Now choose $ρ_{0} > 0$ Mathematical equation small enough such that

$C ρ_{0}^{2} < η \leq c .$ Mathematical equation

Consequently

$J (u) = J (0) = c \geq η > C ρ_{0}^{2} > e^{2} D_{4} = J (0),$ Mathematical equation

which is a contradiction. Therefore, $u$ Mathematical equation is nontrivial critical point.

Following the proof approach of Theorem 1.4 in Ref. [12], we show that $u$ Mathematical equation is a ground state solution.

Take a minimizing sequence ${u_{n}} \subset N$ Mathematical equation , such that

$\underset{n \to \infty}{l i m} J (u_{n}) = c_{0} .$ Mathematical equation (73)

By (73), we may assume (see Ref. [28]) that ${u_{n}}$ Mathematical equation is a Palais-Smale sequence for $J |_{N} .$ Consequently, it follows directly that ${u_{n}}$ is a Palais-Smale sequence for $J$ . Since $J$ satisfies the Palais-Smale condition (Theorem 3), there exists $u_{0} \in E$ such that

$u_{n} \to u_{0} i n E,$ Mathematical equation (74)

combined with the continuity of $J$ Mathematical equation , $J^{'}$ implies that

$J (u_{n}) \to J (u_{0}) = c_{0} a n d J^{'} (u_{0}) u_{0} = \underset{n \to \infty}{l i m} J^{'} (u_{n}) u_{n} = 0,$ Mathematical equation

so $u_{0} \in N$ Mathematical equation .

Consequently, by (73) and (74), we get

$c_{0} \leq J (u_{0}) \leq \underset{n \to \infty}{l i m i n f} J (u_{n}) = c_{0},$ Mathematical equation

then $u_{0}$ Mathematical equation is a ground state solution of system (1).

It is noteworthy that $c = c_{0}$ Mathematical equation (see Lemma 2.8 in Ref. [12]). For the proof concerning the existence of ground state solutions, we also refer the reader to Lemma 2.4 in Ref. [20].

In conclusion, by Lemma 5 and Lemma 6, energy functional $J$ Mathematical equation satisfies the conditions of the Mountain Pass Lemma. By the Palais-Smale condition (Theorem 3), there exists a critical point $u \in E$ , which is a non-trivial solutions of the system, then the ground state solution can be obtained by Theorem 4.

References

Carles R. Remarks on nonlinear Schrödinger equations with harmonic potential[J]. Annales Henri Poincaré, 2002, 3(4): 757-772. [Google Scholar]
Benci V, Fortunato D. Solitons in Schrödinger-Maxwell equations[J]. Journal of Fixed Point Theory and Applications, 2014, 15(1): 101-132. [Google Scholar]
Kaiser H C, Rehberg J. About a one-dimensional stationary Schrödinger-Poisson system with Kohn-Sham potential[J]. Zeitschrift für Angewandte Mathematik und Physik (ZAMP), 1999, 50(3): 423-458. [Google Scholar]
Mauser N J. The Schrödinger-Poisson-Xα equation[J]. Applied Mathematics Letters, 2001, 14(6): 759-763. [Google Scholar]
Ruiz D. The Schrödinger-Poisson equation under the effect of a nonlinear local term[J]. Journal of Functional Analysis, 2006, 237(2): 655-674. [Google Scholar]
Benguria R, Brezis H, Lieb E H. The Thomas-Fermi-von Weizsäcker theory of atoms and molecules[J]. Communications in Mathematical Physics, 1981, 79(2): 167-180. [Google Scholar]
Colin M, Watanabe T. Standing waves for the nonlinear Schrödinger equation coupled with the Maxwell equation[J]. Nonlinearity, 2017, 30(5): 1920-1947. [Google Scholar]
Lieb E H. Thomas-Fermi and related theories of atoms and molecules[J]. Reviews of Modern Physics, 1981, 53(4): 603-641. [Google Scholar]
Jiang Y S, Wang Z P, Zhou H S. Positive solutions for Schrödinger-Poisson-Slater system with coercive potential[J]. Topological Methods in Nonlinear Analysis, 2021, 57: 427-439. [Google Scholar]
Jiang Y S, Zhou H S. Schrödinger-Poisson system with steep potential well[J]. Journal of Differential Equations, 2011, 251(3): 582-608. [Google Scholar]
Jiang Y S, Zhou H S. Schrödinger-Poisson system with singular potential[J]. Journal of Mathematical Analysis and Applications, 2014, 417(1): 411-438. [Google Scholar]
Azzollini A, Pomponio A. Ground state solutions for the nonlinear Schrödinger-Maxwell equations[J]. Journal of Mathematical Analysis and Applications, 2008, 345(1): 90-108. [Google Scholar]
Abdullah Qadha S, Chen H B, Qadha M A. Existence of ground state solutions for choquard equation with the upper critical exponent[J]. Fractal and Fractional, 2023, 7(12): 840. [Google Scholar]
Azzollini A. The planar Schrödinger-Poisson system with a positive potential[J]. Nonlinearity, 2021, 34(8): 5799-5820. [Google Scholar]
Du M. Positive solutions for the Schrödinger-Poisson system with steep potential well[J]. Communications in Contemporary Mathematics, 2023, 25(10): 2250056. [Google Scholar]
Meng Y X, Zhang X R, He X M. Ground state solutions for a class of fractional Schrodinger-Poisson system with critical growth and vanishing potentials[J]. Advances in Nonlinear Analysis, 2020, 10(1): 1328-1355. [Google Scholar]
Peng X Q, Rizzi M. Normalized solutions of mass supercritical Schrödinger-Poisson equation with potential[J]. Calculus of Variations and Partial Differential Equations, 2025, 64(5): 152. [Google Scholar]
Ruiz D, Vaira G. Cluster solutions for the Schrödinger-Poisson-Slater problem around a local minimum of the potential[J]. Revista Matemática Iberoamericana, 2011, 27(1): 253-271. [Google Scholar]
Zhong X X, Zou W M. A new deduction of the strict sub-additive inequality and its application: Ground state normalized solution to Schrödinger equations with potential[J]. Differential and Integral Equations, 2023, 36(1/2):133-160. [Google Scholar]
Shan L Y, Shuai W, Ye J H. Existence of positive solution for Schrödinger-Newton system with a doping profile[J]. Calculus of Variations and Partial Differential Equations, 2025, 65(1): 25. [Google Scholar]
Colin M, Watanabe T. Ground state solutions for Schrödinger-Poisson system with a doping profile[EB/OL]. [2025-10-15]. https://arXivpreprintarXiv:2411.02103. [Google Scholar]
Colin M, Watanabe T. Stable standing waves for nonlinear Schrödinger-Poisson system with a doping profile[J]. Nonlinearity, 2025, 38(11): 115019. [Google Scholar]
Vu N. Mountain pass theorem and nonuniformly elliptic equations[J]. Vietnam J Math, 2005, 33: 391-402. [Google Scholar]
Gordon W B. Physical variational principles which satisfy the Palais-Smale condition[J]. Bulletin of the American Mathematical Society, 1972, 78(5): 712-716. [Google Scholar]
Cazenave T. Semilinear Schrödinger Equations[M]. Providence, Rhode Island: American Mathematical Society, 2003. [Google Scholar]
Ekeland I. Convexity Methods in Hamiltonian Mechanics[M]. Berlin: Springer-Verlag, 1990. [Google Scholar]
Rabinowitz P H. On a class of nonlinear Schrödinger equations[J]. Zeitschrift für Angewandte Mathematik und Physik (ZAMP), 1992, 43(2): 270-291. [Google Scholar]
Willem M. Progress in Nonlinear Differential Equations and Their Applications[M]. Boston: Birkhäuser, 1996. [Google Scholar]

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[R1] Carles R. Remarks on nonlinear Schrödinger equations with harmonic potential[J]. Annales Henri Poincaré, 2002, 3(4): 757-772. [Google Scholar]

[R2] Benci V, Fortunato D. Solitons in Schrödinger-Maxwell equations[J]. Journal of Fixed Point Theory and Applications, 2014, 15(1): 101-132. [Google Scholar]

[R3] Kaiser H C, Rehberg J. About a one-dimensional stationary Schrödinger-Poisson system with Kohn-Sham potential[J]. Zeitschrift für Angewandte Mathematik und Physik (ZAMP), 1999, 50(3): 423-458. [Google Scholar]

[R4] Mauser N J. The Schrödinger-Poisson-Xα equation[J]. Applied Mathematics Letters, 2001, 14(6): 759-763. [Google Scholar]

[R5] Ruiz D. The Schrödinger-Poisson equation under the effect of a nonlinear local term[J]. Journal of Functional Analysis, 2006, 237(2): 655-674. [Google Scholar]

[R6] Benguria R, Brezis H, Lieb E H. The Thomas-Fermi-von Weizsäcker theory of atoms and molecules[J]. Communications in Mathematical Physics, 1981, 79(2): 167-180. [Google Scholar]

[R7] Colin M, Watanabe T. Standing waves for the nonlinear Schrödinger equation coupled with the Maxwell equation[J]. Nonlinearity, 2017, 30(5): 1920-1947. [Google Scholar]

[R8] Lieb E H. Thomas-Fermi and related theories of atoms and molecules[J]. Reviews of Modern Physics, 1981, 53(4): 603-641. [Google Scholar]

[R9] Jiang Y S, Wang Z P, Zhou H S. Positive solutions for Schrödinger-Poisson-Slater system with coercive potential[J]. Topological Methods in Nonlinear Analysis, 2021, 57: 427-439. [Google Scholar]

[R10] Jiang Y S, Zhou H S. Schrödinger-Poisson system with steep potential well[J]. Journal of Differential Equations, 2011, 251(3): 582-608. [Google Scholar]

[R11] Jiang Y S, Zhou H S. Schrödinger-Poisson system with singular potential[J]. Journal of Mathematical Analysis and Applications, 2014, 417(1): 411-438. [Google Scholar]

[R12] Azzollini A, Pomponio A. Ground state solutions for the nonlinear Schrödinger-Maxwell equations[J]. Journal of Mathematical Analysis and Applications, 2008, 345(1): 90-108. [Google Scholar]

[R13] Abdullah Qadha S, Chen H B, Qadha M A. Existence of ground state solutions for choquard equation with the upper critical exponent[J]. Fractal and Fractional, 2023, 7(12): 840. [Google Scholar]

[R14] Azzollini A. The planar Schrödinger-Poisson system with a positive potential[J]. Nonlinearity, 2021, 34(8): 5799-5820. [Google Scholar]

[R15] Du M. Positive solutions for the Schrödinger-Poisson system with steep potential well[J]. Communications in Contemporary Mathematics, 2023, 25(10): 2250056. [Google Scholar]

[R16] Meng Y X, Zhang X R, He X M. Ground state solutions for a class of fractional Schrodinger-Poisson system with critical growth and vanishing potentials[J]. Advances in Nonlinear Analysis, 2020, 10(1): 1328-1355. [Google Scholar]

[R17] Peng X Q, Rizzi M. Normalized solutions of mass supercritical Schrödinger-Poisson equation with potential[J]. Calculus of Variations and Partial Differential Equations, 2025, 64(5): 152. [Google Scholar]

[R18] Ruiz D, Vaira G. Cluster solutions for the Schrödinger-Poisson-Slater problem around a local minimum of the potential[J]. Revista Matemática Iberoamericana, 2011, 27(1): 253-271. [Google Scholar]

[R19] Zhong X X, Zou W M. A new deduction of the strict sub-additive inequality and its application: Ground state normalized solution to Schrödinger equations with potential[J]. Differential and Integral Equations, 2023, 36(1/2):133-160. [Google Scholar]

[R20] Shan L Y, Shuai W, Ye J H. Existence of positive solution for Schrödinger-Newton system with a doping profile[J]. Calculus of Variations and Partial Differential Equations, 2025, 65(1): 25. [Google Scholar]

[R21] Colin M, Watanabe T. Ground state solutions for Schrödinger-Poisson system with a doping profile[EB/OL]. [2025-10-15]. https://arXivpreprintarXiv:2411.02103. [Google Scholar]

[R22] Colin M, Watanabe T. Stable standing waves for nonlinear Schrödinger-Poisson system with a doping profile[J]. Nonlinearity, 2025, 38(11): 115019. [Google Scholar]

[R23] Vu N. Mountain pass theorem and nonuniformly elliptic equations[J]. Vietnam J Math, 2005, 33: 391-402. [Google Scholar]

[R24] Gordon W B. Physical variational principles which satisfy the Palais-Smale condition[J]. Bulletin of the American Mathematical Society, 1972, 78(5): 712-716. [Google Scholar]

[R25] Cazenave T. Semilinear Schrödinger Equations[M]. Providence, Rhode Island: American Mathematical Society, 2003. [Google Scholar]

[R26] Ekeland I. Convexity Methods in Hamiltonian Mechanics[M]. Berlin: Springer-Verlag, 1990. [Google Scholar]

[R27] Rabinowitz P H. On a class of nonlinear Schrödinger equations[J]. Zeitschrift für Angewandte Mathematik und Physik (ZAMP), 1992, 43(2): 270-291. [Google Scholar]

[R28] Willem M. Progress in Nonlinear Differential Equations and Their Applications[M]. Boston: Birkhäuser, 1996. [Google Scholar]