Normalized Solutions for the Choquard Equation With Potential: Concentration and Multiplicity

Abstract

This article focuses on the study of multiplicity and concentration behavior of normalized solutions for a Choquard equation with a local perturbation $\begin{aligned} {\begin{cases} - Δ_{p} u + V (ϵ x) | u |^{p - 2} u = λ | u |^{p - 2} u + (I_{α} * | u |^{q}) | u |^{q - 2} u + μ | u |^{s - 2} u & in R^{N}, \\ \int_{R^{N}} | u |^{p} d x = a^{p} > 0, \end{cases} \end{aligned}$ where $a, ϵ > 0$ , $2 \leq p < N$ , $(p (N + α)) / 2 N < q < (p^{2} + p (N + α)) / 2 N$ , $p < s < p + (p^{2} / N)$ , $μ > 0$ and $λ \in R$ is an unknown parameter that appears as a Lagrange multiplier. Under natural hypotheses, combining the minimization techniques and Ljusternik–Schnirelmann category theory, we obtain the existence and concentration property of normalized solutions for $ϵ > 0$ sufficiently small, as well as the multiplicity result depending on the topology of the set $M$ where the potential $V$ attains its global minimum, which indicates that the numbers of normalized solutions is determined by the topological structure of the set $M$ .

Keywords

normalized solutions multiplicity,concentration Ljusternik–Schnirelmann category

1. Introduction

This article investigates the following $p$ -Laplacian Choquard equation with potential

\begin{aligned} {\begin{cases} - Δ_{p} u + V (ϵ x) | u |^{p - 2} u = λ | u |^{p - 2} u + (I_{α} * | u |^{q}) | u |^{q - 2} u + μ | u |^{s - 2} u & in R^{N}, \\ \int_{R^{N}} | u |^{p} d x = a^{p} > 0, \end{cases} \end{aligned}

(1.1)

where

2 \leq p < N

(p (N + α)) / 2 N < q < (p^{2} + p (N + α)) / 2 N

p < s < p + (p^{2} / N)

μ > 0

. The Riesz potential of order

α \in (0, N)

, denoted by

I_{α} : R^{N} \to R

, is defined at each point

x \in R^{N}

\begin{aligned} I_{α} (x) = \frac{A_{α}}{| x |^{N - α}}, with A_{α} = \frac{Γ (\frac{N - α}{2})}{2^{α} π^{\frac{N}{2}} Γ (\frac{α}{2})} . \end{aligned}

The motivation of this work is to search for normalized solutions to the following nonlinear equation

\begin{aligned} i \frac{\partial ψ}{\partial t} - Δ ψ - γ (I_{α} * | ψ |^{p}) | ψ |^{p - 2} ψ - μ | ψ |^{q - 2} ψ = 0. \end{aligned}

(1.2)

When

p = q = 2

N = 3

α = 2

, and

μ = 0

, (1.2) is called the Choquard–Pekar equation. This model was introduced by Pekar (1954) to describe the quantum theory of a polaron at rest. Later, Choquard Lieb (1977) applied it as an approximation to Hartree–Fock theory of one component plasma. Searching for standing wave solution

ψ (t, x) = e^{- i λ t} u (x)

of (1.2) leads to

\begin{aligned} - Δ u + λ u = γ (I_{α} * | u |^{p}) | u |^{p - 2} u + μ | u |^{q - 2} u . \end{aligned}

(1.3)

There exist two substantially different view of points in terms of the frequency

λ

in (1.3). One is to regard

λ

as a given constant. In this situation,

u

solves (1.3) iff it is a critical point of the corresponding energy functional

\begin{aligned} E (u) := \frac{1}{2} \int_{R^{N}} | \nabla u |^{2} + λ | u |^{2} d x - \frac{γ}{2 p} \int_{R^{N}} (I_{α} * | u |^{p}) | u |^{p} d x - \frac{μ}{q} \int_{R^{N}} | u |^{q} d x . \end{aligned}

The other one is to regard

λ

as unknown quantities to (1.3). In this case, it is natural to prescribe the valve of the mass

\int_{R^{N}} | u |^{2} d x

so that

λ

can be interpreted as a Lagrange multiplier. From a physical point of view, it is interesting to seek solutions satisfying

\int_{R^{N}} | u |^{2} d x = a^{2}

for a given

a

. Such solutions are called normalized solutions. A normalized solution of (1.3) corresponds to a critical point of the functional

\begin{aligned} I (u) := \frac{1}{2} \int_{R^{N}} | \nabla u |^{2} d x - \frac{γ}{2 p} \int_{R^{N}} (I_{α} * | u |^{p}) | u |^{p} d x - \frac{μ}{q} \int_{R^{N}} | u |^{q} d x . \end{aligned}

restricted to the sphere

\begin{aligned} S_{a} := {u \in H^{1} (R^{N}) : \int_{R^{N}} | u |^{2} d x = a^{2}} . \end{aligned}

Recently, normalized solutions of Choquard equation (1.3) have attracted much attention. Jeanjean and Le (2021) considered the existence and multiplicity of the normalized solutions of (1.3) in the case

N = 3

. They proved (1.3) admits positive and negative solutions when

q \in (10 / 3, 6]

γ > 0

and

μ > 0

. If

γ > 0

and

μ < 0

, (1.3) has a ground state solution. For a

L^{2}

-critical and

L^{2}

-supercritical perturbation

μ | u |^{q - 2} u

, Li (2023) studied the nonexistence, existence, and symmetry of normalized ground states for (1.3). For more results, we can refer the readers to Alreshidi and Hai (2024), Bartsch et al. (2020), Bonanno et al. (2014), Li et al. (2022, 2023), Lions (1980), Mo and Ma (2025), Papageorgiou et al. (2024), Papageorgiou, Zhang, et al. (2025)), Tripathi (2025), Yao et al. (2022), Ye (2025), Zhang and Zhang (2025) and the references therein.

In a more general framework, we will consider the normalized solutions of (1.1) with potential $V$ . Under appropriate assumptions about $V$ , Ye (2016) obtained the existence of normalized ground states to the following equation

\begin{aligned} {\begin{cases} - Δ u + V (ϵ x) u = λ u + (I_{α} * | u |^{p}) | u |^{p - 2} u in R^{N}, \\ \int_{R^{N}} | u |^{2} d x = a^{2} > 0, \end{cases} \end{aligned}

(1.4)

where

((N + α) / N) \leq p < ((N + α) / (N - 2))

. Moroz and Schaftingen (2015) also studied the same equation; they proved (1.4) has a ground state and a family of solutions concentrating to the local minimum of

V

for

2 \leq p < ((N + α) / (N - 2))

For the case $p \neq 2$ , there is little literature on the existence of normalized solutions for the $p$ -Laplacian equation. Wang and Sun (2023) considered the following $p$ -Laplacian equation with trapping potential

\begin{aligned} {\begin{cases} - Δ_{p} u + V (x) | u |^{p - 2} u = λ | u |^{p - 2} u + | u |^{q - 2} u in R^{N}, \\ \int_{R^{N}} | u |^{r} d x = a, \end{cases} \end{aligned}

(1.5)

where

r = p

or 2. In the case

r = p

, they obtained a ground state solution with positive energy for

p + (p^{2}) / N < q < p^{*}

. When

r = 2

, problem (1.5) admits at least one ground state and one high-energy solution. Gu et al. (2017) proved the existence of ground states of (1.5) for

q = p + (p^{2}) / N

. Lou and Zhang (2024) studied the following

p

-Laplacian equation with the

L^{p}

-mass constraint

\begin{aligned} {\begin{cases} - Δ_{p} u = λ | u |^{p - 2} u + | u |^{q - 2} u & x \in R^{N}, \\ \int_{R^{N}} | u |^{p} d x = a^{p} > 0, \end{cases} \end{aligned}

where

1 < p < q < p^{*}

. Firstly, they established the existence of normalized solutions for the

p

-Laplacian equation and provided precise characterizations of these solutions. Subsequently, the work was extended to problem (1.5). Under different assumptions on the exponent

q

and the mass

a

, they obtained the existence, nonexistence, concentration phenomenon and exponential decay of normalized solutions of (1.5). The same equation is considered by Deng and Wu (2023). They established the existence of a mountain pass solution possessing positive energy. For more results involving the

p

-Laplacian problem, we can refer the readers to Papageorgiou et al. (2022), Papageorgiou, Rădulescu, et al. (2025), Shen and Squassina (2025), Zhang et al. (2022), and Zhang and Zhang (2022).

Alves and Nguyen (2023) considered the following equation

\begin{aligned} {\begin{cases} - Δ u + V (ϵ x) u = λ u + f (u) & in R^{N}, \\ \int_{R^{N}} | u |^{2} d x = a^{2} > 0, \end{cases} \end{aligned}

(1.6)

where

f

is odd and

L^{2}

-subcritical, and the function

V

satisfies

\begin{aligned} 0 = inf_{x \in R^{N}} V (x) < \underset{| x | \to + \infty}{lim inf} V (x) = V_{\infty} . \end{aligned}

They obtained a connection between the number of normalized solutions and the topological structure of the set where the potential

V

reaches its minimum. The proof of the main results employs the minimization methods combined with the Lusternik–Schnirelmann category theory. Very recently, a penalization approach was applied by Alves and Nguyen (2024) to handle the case that the potential

V

satisfies a local condition. We also refer to Li, Nie, et al. (2025) and Li, Rădulescu, et al. (2025) for more results about the multiplicity and concentration of normalized solutions to fractional Schrödinger equations.

Inspired by the above facts, in the present article, we are interested in the qualitative and asymptotic analysis of normalized solutions to problem (1.1), and we are mainly concerned with existence, multiplicity properties of normalized solutions, as well as with concentration phenomenon as $ϵ \to 0$ . Here the Choquard reaction term of (1.1) has subcritical growth related to the Hardy–Littlewood–Sobolev exponent, and the perturbation term $μ | u |^{s - 2} u$ also satisfies subcritical growth condition. To the best of our knowledge, it seems that such a problem was not considered in the literature before.

In this article, we define the set

\begin{aligned} S_{a} = {u \in W^{1, p} (R^{N}) : \int_{R^{N}} | u |^{p} d x = a^{p}} . \end{aligned}

Since we are going to deal with the nonlocal type problem (1.1), we would like to recall the classical Hardy–Littlewood–Sobolev inequality, which is crucial for our subsequent proofs.

Lemma 1.1

Suppose $α \in (0, N)$ , and $t, r > 1$ with $(1 / r) + (1 / t) = 1 + (α / N)$ . Let $f \in L^{t} (R^{N})$ and $g \in L^{r} (R^{N})$ , then there exists a constant $C (N, α, t, r) > 0$ such that

\begin{aligned} | \int_{R^{N}} \int_{R^{N}} \frac{f (x) g (y)}{| x - y |^{N - α}} d x d y | \leq C (N, α, t, r) | f |_{t} | g |_{r} . \end{aligned}

By applying the Lemma 1.1, we deduce that $q$ satisfies the inequality

\begin{aligned} \frac{p (N + α)}{2 N} < q < \frac{p (N + α)}{2 (N - p)} . \end{aligned}

Here the exponent

(p (N + α)) / 2 N

is called the lower critical exponent and

(p (N + α)) / (2 (N - p))

is called the upper critical exponent. In this article, we only consider the subcritical case, hence the inequality holds with a strict sign. Since this article focuses on the normalized solution of problem (1.1), we now introduce another key inequality: Gagliardo–Nirenberg inequality.

Lemma 1.2 Gagliardo–Nirenberg inequality

Let $r \in (p, p^{*})$ . Then there exists a sharp constant $C_{N, r} > 0$ such that

\begin{aligned} ‖ u ‖_{r} \leq C_{N, r} ‖ \nabla u ‖_{p}^{δ_{r}} ‖ u ‖_{p}^{1 - δ_{r}}, \forall u \in W^{1, p} (R^{N}), \end{aligned}

where

δ_{r} := (N / p) - (N / r)

For the potential $V$ , we assume that it satisfies the following assumption:

( $V$ )
$V \in C (R^{N}, R) \cap L^{\infty} (R^{N})$ and
$\begin{aligned} 0 = V_{0} := inf_{x \in R^{N}} V (x) < V_{\infty} := \underset{| x | \to + \infty}{lim inf} V (x) . \end{aligned}$

To describe the multiplicity result of normalized solutions of problem (1.1), we need use the Ljusternik–Schnirelman category theory. Now let us remind the definition of the Ljusternik–Schnirelman category. Let $X$ be a topological space and let $Y \neq \emptyset$ be a closed subset of $X$ . The category of $Y$ in $X$ , $c a t_{X} (Y)$ , is the smallest integer $n$ such that
$\begin{aligned} Y \subset ⋃_{k = 1}^{n} A_{k}, \end{aligned}$
where for each $k = 1, \cdot, \cdot, \cdot, n$ , $A_{k}$ is a closed set contractible in $X$ . If such a integer does not exist, then $c a t_{X} (Y) = + \infty$ . In this context, the category of $X$ in itself, $c a t_{X} (X)$ , is simply denoted by $c a t (X)$ . Moreover, we define two sets
$\begin{aligned} M = {x \in R^{N} : V (x) = V_{0}} \end{aligned}$
and
$\begin{aligned} M_{δ} = {x \in R^{N} : dist (x, M) \leq δ} . \end{aligned}$

We state in what follows the main results of this article.
Theorem 1.1
Let $p (N + α) / 2 N < q < (p^{2} + p (N + α)) / 2 N$ and $p < s < p + (p^{2}) / N$ . Then for each $δ > 0$ , there is a constant $ϵ_{0} > 0$ and $V_{} > 0$ such that (1.1) has at least $c a t_{M_{δ}} (M)$ couples $(u_{j}, λ_{j}) \in W^{1, p} (R^{N}) \times R$ of weak solutions for $ϵ \in (0, ϵ_{0})$ and $| V |_{\infty} < V_{}$ with $λ_{j} < 0$ , and $J_{ϵ} (u_{j}) < 0$ . Moreover, if $u_{ϵ}$ denotes one of these solutions and $ξ_{ϵ}$ is the global maximum of $| u_{ϵ} |$ , then
$\begin{aligned} lim_{ϵ \to 0} V (ϵ ξ_{ϵ}) = V_{0} . \end{aligned}$

The article is divided into four sections. In Section 2, we study the autonomous problem. In Section 3, we investigates the non-autonomous case, including an analysis of the Palais–Smale condition on $S_{a}$ for the energy functional, along with key results for establishing multiplicity result. In Section 4, we present the proof of the multiplicity and concentration of normalized solutions to problem (1.1).

Throughout this article, for the sake of simplicity we will use the following notations.

$‖ \cdot ‖_{s}$ denotes the usual norm of the space $L^{s} (R^{N})$ , $1 \leq s \leq \infty$ ;

$‖ \cdot ‖$ denotes the norm of the Sobolev space $W^{1, p} (R^{N})$ ;

$c$ , $C$ , $c_{i}$ , $C_{i}$ denote some different positive constants;

for any $x \in R^{N}$ and $r > 0$ , $B_{r} (x) := {y \in Ω : | y - x | < r}$ .

2. The Autonomous Case

In this section, we investigate the existence of normalized solutions to the following autonomous problem

\begin{aligned} {\begin{cases} - Δ_{p} u + ν | u |^{p - 2} u = λ | u |^{p - 2} u + (I_{α} * | u |^{q}) | u |^{q - 2} u + μ | u |^{s - 2} u & in R^{N}, \\ \int_{R^{N}} | u |^{p} d x = a^{p} > 0, \end{cases} \end{aligned}

(2.1)

where

ν \geq 0

. The energy functional of problem (2.1) is defined by

\begin{aligned} J_{ν} (u) = \frac{1}{p} \int_{R^{N}} | \nabla u |^{p} + ν | u |^{p} d x - \frac{1}{2 q} \int_{R^{N}} (I_{α} * | u |^{q}) | u |^{q} d x - \frac{μ}{s} \int_{R^{N}} | u |^{s} d x . \end{aligned}

Lemma 2.1

The functional $J_{ν}$ is coercive and bounded from below on $S_{a}$ .

Proof.

From Lemmas 1.1 and 1.2, we have

\begin{aligned} J_{ν} (u) & = \frac{1}{p} \int_{R^{N}} | \nabla u |^{p} + ν | u |^{p} d x - \frac{1}{2 q} \int_{R^{N}} (I_{α} * | u |^{q}) | u |^{q} d x - \frac{μ}{s} \int_{R^{N}} | u |^{s} d x \\ \geq \frac{1}{p} ‖ \nabla u ‖_{p}^{p} + \frac{ν}{p} ‖ u ‖_{p}^{p} - \frac{1}{2 q} ‖ u ‖_{\frac{2 N q}{N + α}}^{2 q} - \frac{μ}{s} ‖ u ‖_{s}^{s} \\ \geq \frac{1}{p} ‖ \nabla u ‖_{p}^{p} - C_{1} a^{(1 - \frac{N}{p} + \frac{N + α}{2 q}) 2 q} ‖ \nabla u ‖_{p}^{2 q (\frac{N}{p} - \frac{N + α}{2 q})} - C_{2} a^{(1 - δ_{s}) s} ‖ \nabla u ‖_{p}^{s δ_{s}} . \end{aligned}

The inequalities

2 q (N / p - (N + α / 2 q)) < p

and

s δ_{s} < p

imply that

J_{ν}

is coercive and bounded from below on

S_{a}

The above lemma shows that

\begin{aligned} E_{ν, a} = inf_{u \in S_{a}} J_{ν} (u) \end{aligned}

is well defined. Next, we give some useful results associated with

E_{ν, a}

Lemma 2.2

There exists $V_{*} > 0$ such that $E_{ν, a} < 0$ for $0 \leq ν < V_{*}$ .

Proof.

Let $u \in S_{a}$ be fixed, then we have $u_{t} (x) = t^{(N / p)} u (t x) \in S_{a}$ and

\begin{aligned} J_{ν} (u_{t}) & = \frac{t^{p}}{p} ‖ \nabla u ‖_{p}^{p} + \frac{ν}{p} a^{p} - \frac{1}{2 q} \int_{R^{N}} (I_{α} * | u_{t} |^{q}) | u_{t} |^{q} d x - \frac{t^{s δ_{s}}}{s} ‖ u ‖_{s}^{s} \\ \leq \frac{t^{p}}{p} ‖ \nabla u ‖_{p}^{p} + \frac{ν}{p} a^{p} - \frac{t^{s δ_{s}}}{s} ‖ u ‖_{s}^{s} . \end{aligned}

For

t > 0

sufficiently small, we have

\begin{aligned} \frac{t^{p}}{p} ‖ \nabla u ‖_{p}^{p} - \frac{t^{s δ_{s}}}{s} ‖ u ‖_{s}^{s} = A_{t} < 0. \end{aligned}

Therefore, there holds

\begin{aligned} J_{ν} \leq A_{t} + \frac{ν}{p} a^{p} . \end{aligned}

We can choose a suitable

V_{*} > 0

such that

\begin{aligned} A_{t} + \frac{V_{*}}{p} a^{p} < 0. \end{aligned}

Consequently, for all

ν \in [0, V_{*})

, if

ν < V_{*}

, then

\begin{aligned} J_{ν} (u_{t}) < 0. \end{aligned}

As a consequence, we have

E_{ν, a} < J_{ν} (u_{t}) < 0

Lemma 2.3

Fix $ν \in [0, V_{*})$ and let $0 < a_{1} < a_{2}$ . Then $a_{1}^{p} E_{ν, a_{2}} < a_{2}^{p} E_{ν, a_{1}} < 0$ .

Proof.

Let $θ = (a_{2} / a_{1})$ and ${u_{n}} \subset S_{a_{1}}$ is a minimizing sequence for $J_{ν, a_{1}}$ . Considering $v_{n} := θ u_{n} \in S_{a_{2}}$ , then we have

\begin{aligned} E_{ν, a_{2}} & \leq J_{ν, a_{2}} = \frac{1}{p} ‖ \nabla v_{n} ‖_{p}^{p} + \frac{ν}{p} a_{2}^{p} - \frac{1}{2 q} \int_{R^{N}} (I_{α} * | v_{n} |^{q}) | v_{n} |^{q} d x - \frac{μ}{s} ‖ v_{n} ‖_{s}^{s} \\ = \frac{1}{p} θ^{p} ‖ \nabla u_{n} ‖_{p}^{p} + \frac{ν}{p} θ^{p} a_{1}^{p} - \frac{θ^{2 q}}{2 q} \int_{R^{N}} (I_{α} * | u_{n} |^{q}) | u_{n} |^{q} d x - \frac{μ}{s} θ^{s} ‖ u_{n} ‖_{s}^{s} \\ = θ^{p} J_{ν, a_{1}} + \frac{θ^{p} - θ^{2 q}}{2 q} \int_{R^{N}} (I_{α} * | u_{n} |^{q}) | u_{n} |^{q} d x + \frac{μ (θ^{p} - θ^{s})}{s} ‖ u_{n} ‖_{s}^{s} . \end{aligned}

Since

p < 2 q

p < s

and

θ > 1

, it follows that

\begin{aligned} E_{ν, a_{2}} < θ^{p} J_{ν, a_{1}} . \end{aligned}

Taking the limit as

n \to \infty

, we obtain

\begin{aligned} E_{ν, a_{2}} < θ^{p} E_{ν, a_{1}}, \end{aligned}

which implies that

a_{1}^{p} E_{ν, a_{2}} < a_{2}^{p} E_{ν, a_{1}}

In what follows, we prove the compactness lemma on $S_{a}$ , which is useful in the nonautonomous case.

Lemma 2.4

Let $ν \in [0, V_{*})$ and ${u_{n}} \subset S_{a}$ be a minimizing sequence of $E_{ν, a}$ . Then one of the following conclusions holds: (i)

${u_{n}}$ has a strongly convergence subsequence in $W^{1, p} (R^{N})$ ;

(ii)

there exists a sequence ${y_{n}} \subset R^{N}$ with $| y_{n} | \to + \infty$ such that the sequence ${w_{n} := u_{n} (x + y_{n})}$ is strongly convergent to a function $w \in S_{a}$ with $J_{μ} (w) = E_{μ, a}$ .

Proof.

Let ${u_{n}} \subset S_{a}$ be a minimizing sequence of $E_{ν, a}$ , then ${u_{n}}$ satisfies

\begin{aligned} J_{ν} (u_{n}) \to E_{ν, a} and J_{ν}^{'} (u_{n}) \to 0. \end{aligned}

From Lemma 2.1, we can clearly see that

{u_{n}}

is bounded. Then, up to a subsequence, we can assume that

u_{n} ⇀ u

W^{1, p} (R^{N})

. We next continue our arguments by distinguishing two cases.

Case 1: $u \neq 0$ . We first prove that $\nabla u_{n} \to \nabla u$ a.e. on $R^{N}$ . Our proof is based on the arguments in Zhang et al. (2023). Let $φ \in C_{0}^{\infty} (R^{N}, [0, 1])$ , where $φ |_{B_{R} (0)} = 1$ , and $supp φ \subset B_{2 R} (0)$ . Computing directly,

\begin{aligned} \int_{R^{N}} (| \nabla u_{n} |^{p - 2} \nabla u_{n} - | \nabla u |^{p - 2} \nabla u) (\nabla u_{n} - \nabla u) φ d x \\ = ⟨ J_{ν}^{'} (u_{n}), (u_{n} - u) φ ⟩ - \int_{R^{N}} | \nabla u_{n} |^{p - 2} \nabla u_{n} (u_{n} - u) \nabla φ d x - \int_{R^{N}} | \nabla u |^{p - 2} \nabla u (\nabla u_{n} - \nabla u) φ d x \\ - ν \int_{R^{N}} | u_{n} |^{p - 2} u_{n} (u_{n} - u) φ d x + μ \int_{R^{N}} | u_{n} |^{s - 2} u_{n} (u_{n} - u) φ d x \\ + \int_{R^{N}} (I_{α} * | u_{n} |^{q}) | u_{n} |^{q - 2} u_{n} (u_{n} - u) φ d x . \end{aligned}

Since

{u_{n}} \subset S_{a}

is bounded and

J_{ν}^{'} (u_{n}) \to 0

, then we have

\begin{aligned} ⟨ J_{ν}^{'} (u_{n}), (u_{n} - u) φ ⟩ = o_{n} (1) . \end{aligned}

(2.2)

From the compact embedding

W^{1, p} (R^{N}) ↪ L_{l o c}^{r} (R^{N})

for

1 \leq r < p^{*}

, and

u_{n} ⇀ u

W^{1, p} (R^{N})

, we deduce

\begin{aligned} \begin{aligned} \int_{R^{N}} | \nabla u_{n} |^{p - 2} \nabla u_{n} (u_{n} - u) \nabla φ d x = o_{n} (1), \\ \int_{R^{N}} | \nabla u |^{p - 2} \nabla u (\nabla u_{n} - \nabla u) φ d x = o_{n} (1), \\ \int_{R^{N}} | u_{n} |^{p - 2} u_{n} (u_{n} - u) φ d x = o_{n} (1), \\ \int_{R^{N}} | u_{n} |^{s - 2} u_{n} (u_{n} - u) φ d x = o_{n} (1) . \end{aligned} \end{aligned}

(2.3)

By Lemma 1.1, we can infer that

\begin{aligned} \int_{R^{N}} (I_{α} * | u_{n} |^{q}) | u_{n} |^{q - 2} u_{n} (u_{n} - u) φ d x = o_{n} (1) . \end{aligned}

(2.4)

Combining (2.2)–(2.4), we get

\begin{aligned} \int_{R^{N}} (| \nabla u_{n} |^{p - 2} \nabla u_{n} - | \nabla u |^{p - 2} \nabla u) (\nabla u_{n} - \nabla u) φ d x = o_{n} (1) . \end{aligned}

Moreover, applying the following inequality

\begin{aligned} ⟨ | ξ |^{s - 2} ξ - | η |^{s - 2} η, ξ - η ⟩ \geq {\begin{cases} c | ξ - η |^{s}, & if s \geq 2, \\ c (| ξ | + | η |)^{s - 2} | ξ - η |^{2}, & if 1 < s < 2, \end{cases} \forall ξ, η \in R^{N} \end{aligned}

we can see that

\begin{aligned} \nabla u_{n} \to \nabla u a.e. x \in R^{N} . \end{aligned}

Set $v_{n} = u_{n} - u$ . Using the above fact and Brézis–Lieb lemma, we can derive that

\begin{aligned} ‖ \nabla u_{n} ‖_{p}^{p} & = ‖ \nabla v_{n} ‖_{p}^{p} + ‖ \nabla u ‖_{p}^{p} + o_{n} (1), \end{aligned}

(2.5)

\begin{aligned} ‖ u_{n} ‖_{p}^{p} & = ‖ v_{n} ‖_{p}^{p} + ‖ u ‖_{p}^{p} + o_{n} (1), \end{aligned}

(2.6)

\begin{aligned} ‖ u_{n} ‖_{s}^{s} & = ‖ v_{n} ‖_{s}^{s} + ‖ u ‖_{s}^{s} + o_{n} (1) . \end{aligned}

(2.7)

Moreover, employing (Zhang et al., 2023, Lemma 4.1) we can obtain

\begin{aligned} \int_{R^{N}} (I_{α} * | u_{n} |^{q}) | u_{n} |^{q} d x = \int_{R^{N}} (I_{α} * | v_{n} |^{q}) | v_{n} |^{q} d x + \int_{R^{N}} (I_{α} * | u |^{q}) | u |^{q} d x + o_{n} (1) . \end{aligned}

(2.8)

In what follows, we demonstrate that $‖ v_{n} ‖_{p} \to 0$ . Set $‖ v_{n} ‖_{p} = d_{n}$ , $‖ u ‖_{p} = b$ , and we suppose $‖ v_{n} ‖_{p} \to d > 0$ , then $b < a$ . From (2.5)–(2.8) and Lemma 2.3, we deduce that

\begin{aligned} E_{ν, a} + o_{n} (1) & = J_{ν} (u_{n}) + o_{n} (1) = J_{ν} (v_{n}) + J_{ν} (u) + o_{n} (1) \\ \geq E_{ν, d_{n}} + E_{ν, b} + o_{n} (1) \\ \geq \frac{d_{n}^{p}}{a^{p}} E_{ν, a} + E_{ν, b} + o_{n} (1) \end{aligned}

Letting

n \to \infty

, we can see that

\begin{aligned} E_{ν, a} \geq & \frac{d^{p}}{a^{p}} E_{ν, a} + E_{ν, b} \\ > & \frac{d^{p}}{a^{p}} E_{ν, a} + \frac{b^{p}}{a^{p}} E_{ν, a} = E_{ν, a} . \end{aligned}

Clearly, this leads to a contradiction. Using this fact and Lemma 1.2, we derive that

\begin{aligned} ‖ v_{n} ‖_{s} \leq C_{1} ‖ \nabla v_{n} ‖_{p}^{δ_{s}} ‖ v_{n} ‖_{p}^{1 - δ_{s}} \to 0. \end{aligned}

(2.9)

By Lemma 1.1, we have

\begin{aligned} \int_{R^{N}} (I_{α} * | v_{n} |^{q}) | v_{n} |^{q} d x \leq ‖ v_{n} ‖_{\frac{2 N q}{N + α}}^{2 q} \leq C_{2} ‖ v_{n} ‖_{p}^{(1 - \frac{N}{p} + \frac{N + α}{2 q}) 2 q} ‖ \nabla v_{n} ‖_{p}^{2 q (\frac{N}{p} - \frac{N + α}{2 q})} \to 0. \end{aligned}

(2.10)

Combining (2.9) and (2.10), we obtain

\begin{aligned} E_{ν, a} & = J_{ν} (v_{n}) + J_{ν} (u) + o_{n} (1) \\ \geq J_{ν} (v_{n}) + E_{ν, a} + o_{n} (1) \\ \geq \frac{1}{p} ‖ \nabla v_{n} ‖_{p}^{p} + E_{ν, a} + o_{n} (1) . \end{aligned}

This indicates that

‖ \nabla v_{n} ‖_{p} \to 0

, which shows that

u_{n} \to u

W^{1, p} (R^{N})

Case 2: $u = 0$ . We first claim that

\begin{aligned} lim_{n \to \infty} sup_{y \in R^{N}} \int_{B_{R} (y)} | u_{n} |^{p} d x > 0. \end{aligned}

If not, using Lion’s concentration compactness principle in Lions (1984), we have

‖ u_{n} ‖_{r} \to 0

for

p < r < p^{*}

. Thus, by Lemma 1.1 we can obtain

\begin{aligned} 0 & > E_{ν, a} = lim_{n \to \infty} J_{ν} (u_{n}) \\ = lim_{n \to \infty} [\frac{1}{p} ‖ \nabla u_{n} ‖_{p}^{p} + \frac{ν}{p} a^{p} - \frac{1}{2 q} \int_{R^{N}} (I_{α} * | u_{n} |^{q}) | u_{n} |^{q} d x - \frac{μ}{s} ‖ u_{n} ‖_{s}^{s}] \\ \geq lim_{n \to \infty} [\frac{1}{p} ‖ \nabla u_{n} ‖_{p}^{p} + \frac{ν}{p} a^{p} - \frac{1}{2 q} ‖ u_{n} ‖_{\frac{2 N q}{N + α}}^{2 q} - \frac{μ}{s} ‖ u_{n} ‖_{s}^{s}] \\ \geq 0. \end{aligned}

Evidently, we get a contradiction. Therefore, there exist

R, σ_{0}

and

{y_{n}} \subset R^{N}

such that

\begin{aligned} \int_{B_{R} (y_{n})} | u_{n} (x) |^{p} d x \geq σ_{0} > 0. \end{aligned}

Moreover, we can know that

{y_{n}}

is unbounded in

R^{N}

. Set

w_{n} := u_{n} (x + y_{n})

, then,

w_{n} \in S_{a}

is also a minimizing sequence of

E_{ν, a}

. Therefore, there exists

w \in W^{1, p} (R^{N})

such that

w_{n} ⇀ w

W^{1, p} (R^{N})

and

w_{n} (x) \to w (x)

a.e. on

R^{N}

. The remaining proofs are similar to the proofs of Case 1, here we omit the details, and we complete the proof of the lemma.

Lemma 2.5

Let $ν \in [0, V_{*})$ , then problem (2.1) has solutions $(u, λ)$ , where $u$ is positive and radially symmetric, and $λ < 0$ .

Proof.

Applying Lemma 2.4, we can see that $E_{ν, a}$ is attained by some $u$ . The Lagrange multiplier rule implies that there exists $λ_{a} \in R$ such that

\begin{aligned} J_{ν}^{'} (u) = λ_{a} ψ^{'} (u), \end{aligned}

where

\begin{aligned} ψ (u) = \int_{R^{N}} | u |^{p} d x, u \in W^{1, p} (R^{N}) . \end{aligned}

Therefore,

\begin{aligned} - Δ_{p} u + ν | u |^{p - 2} u = λ_{a} | u |^{p - 2} u + (I_{α} * | u |^{q}) | u |^{q - 2} u + μ | u |^{s - 2} u in R^{N} . \end{aligned}

According to Lemma 2.2, we obtain

\begin{aligned} 0 & > E_{ν, a} = J_{ν} (u) \\ = \frac{1}{p} \int_{R^{N}} | \nabla u |^{p} + ν | u |^{p} d x - \frac{1}{2 q} \int_{R^{N}} (I_{α} * | u |^{q}) | u |^{q} d x - \frac{μ}{s} \int_{R^{N}} | u |^{s} d x \\ = \frac{λ_{a}}{p} \int_{R^{N}} | u |^{p} d x + \frac{1}{p} \int_{R^{N}} (I_{α} * | u |^{q}) | u |^{q} d x + \frac{μ}{p} \int_{R^{N}} | u |^{s} d x - \frac{μ}{s} \int_{R^{N}} | u |^{s} d x - \frac{1}{2 q} \int_{R^{N}} (I_{α} * | u |^{q}) | u |^{q} d x \\ = (\frac{1}{p} - \frac{1}{2 q}) \int_{R^{N}} (I_{α} * | u |^{q}) | u |^{q} d x + (\frac{μ}{p} - \frac{μ}{s}) \int_{R^{N}} | u |^{s} d x + \frac{λ_{a}}{p} \int_{R^{N}} | u |^{p} d x . \end{aligned}

So, we can infer that

λ_{a} < 0

We now establish the positivity and radial symmetry of $u$ . According to the definition of functional $J_{ν}$ , we can easily check $J_{ν} (| u |) = J_{ν} (u)$ . Moreover, since $u \in S_{a}$ implies $| u | \in S_{a}$ , we get

\begin{aligned} E_{ν, a} \leq J_{ν} (| u |) = J_{ν} (u) = E_{ν, a}, \end{aligned}

(2.11)

which implies that

J_{ν} (| u |) = E_{ν, a}

. Thereby, we can replace

u

| u |

. Furthermore, if

u^{*}

denotes the Schwarz symmetrization of

u

, we know that

\begin{aligned} \int_{R^{N}} | \nabla u |^{p} d x \geq \int_{R^{N}} | \nabla u^{*} |^{p} d x, \int_{R^{N}} | u |^{p} d x = \int_{R^{N}} | u^{*} |^{p} d x \end{aligned}

and

\begin{aligned} \int_{R^{N}} (I_{α} * | u |^{q}) | u |^{q} d x = \int_{R^{N}} (I_{α} * | u^{*} |^{q}) | u^{*} |^{q} d x, \int_{R^{N}} | u |^{s} d x = \int_{R^{N}} | u^{*} |^{s} d x . \end{aligned}

Then,

u^{*} \in S_{a}

and

\begin{aligned} E_{ν, a} \leq J_{ν} (u^{*}) \leq J_{ν} (u) = E_{ν, a}, \end{aligned}

that is,

J_{ν} (u^{*}) = E_{ν, a}

. In other words, we can replace

u

u^{*}

. Finally, applying the strong maximum principle (Pucci & Serrin, 2000), we can know that

u

is positive. The proof is complete.

Remark 2.1

According to the proof of Lemma 2.5, we know that $u$ satisfies $J_{ν} (u) = E_{ν, a}$ .

The following corollary is a product of Lemma 2.5 which will be used in the later.

Corollary 2.1

Fix $a > 0$ , let $0 < ν_{1} < ν_{2} < V_{*}$ . Then, $E_{ν_{1}, a} < E_{ν_{2}, a} < 0$ .

Proof.

Take $u_{ν_{2}, a} \in S_{a}$ such that $J_{ν_{2}} (u_{ν_{2}, a}) = E_{ν_{2}, a}$ . Then,

\begin{aligned} E_{ν_{1}, a} \leq J_{ν_{1}} (u_{ν_{2}, a}) < J_{ν_{2}} (u_{ν_{2}, a}) = E_{ν_{2}, a} < 0. \end{aligned}

3. The Nonautonomous Case

In this section, we always assume that $‖ V ‖_{\infty} \leq V_{*}$ . We define the energy functional associated with problem (1.1)

\begin{aligned} J_{ϵ} (u) = \frac{1}{p} \int_{R^{N}} | \nabla u |^{p} + V (ϵ x) | u |^{p} d x - \frac{1}{2 q} \int_{R^{N}} (I_{α} * | u |^{q}) | u |^{q} d x - \frac{μ}{s} \int_{R^{N}} | u |^{s} d x . \end{aligned}

Let

\begin{aligned} E_{ϵ, a} = inf_{u \in S_{a}} J_{ϵ} (u), E_{0, a} = inf_{u \in S_{a}} J_{0} (u), E_{\infty, a} = inf_{u \in S_{a}} J_{\infty} (u), \end{aligned}

where

J_{\infty} (\cdot) = J_{V_{\infty}} (\cdot)

. Then, according to

(V)

and Corollary 2.1, we can obtain

\begin{aligned} E_{0, a} \leq E_{\infty, a} < 0. \end{aligned}

(3.1)

Next we establish the relations between $E_{ϵ, a}$ , $E_{0, a}$ and $E_{\infty, a}$ .

Lemma 3.1

$\underset{ϵ \to 0^{+}}{lim sup} E_{ϵ, a} \leq E_{0, a}$ , and there exists $ϵ_{0} > 0$ such that $E_{ϵ, a} < E_{\infty, a}$ for all $ϵ \in (0, ϵ_{0})$ .

Proof.

Let $u_{0} \in S_{a}$ be a minimizer of $J_{0} (u)$ , satisfying $J_{0} (u_{0}) = E_{0, a}$ . Thus,

\begin{aligned} \underset{ϵ \to 0^{+}}{lim sup} E_{ϵ, a} & \leq \underset{ϵ \to 0^{+}}{lim sup} J_{ϵ, a} (u_{0}) \\ = \underset{ϵ \to 0^{+}}{lim sup} {\frac{1}{p} \int_{R^{N}} | \nabla u_{0} |^{p} + V (ϵ x) | u_{0} |^{p} d x - \frac{1}{2 q} \int_{R^{N}} (I_{α} * | u_{0} |^{q}) | u_{0} |^{q} d x - \frac{μ}{s} \int_{R^{N}} | u_{0} |^{s} d x} . \\ = J_{0, a} (u_{0}) = E_{0, a} . \end{aligned}

The remaining conclusion follows directly from (3.1).

In the subsequent lemmas, we set $ρ_{1} = (1 / 2) (E_{\infty, a} - E_{0, a})$ with $ρ_{1} > 0$ .

Lemma 3.2

Fix $ϵ \in (0, ϵ_{0})$ and let ${u_{n}} \subset S_{a}$ such that $J_{ϵ} (u_{n}) \to c$ with $c < E_{0, a} + ρ_{1} < 0$ . If $u_{n} ⇀ u$ in $W^{1, p} (R^{N})$ , then $u \neq 0$ .

Proof.

Assume by contradiction that $u = 0$ . From $(V)$ , for any given $ζ > 0$ , there exists $R > 0$ such that

\begin{aligned} V (x) \geq V_{\infty} - ζ \forall x \geq R . \end{aligned}

By Lemma 2.1, we know that

{u_{n}}

is bounded. Hence,

\begin{aligned} E_{0, a} + ρ_{1} + o_{n} (1) & > c + o_{n} (1) = J_{ϵ} (u_{n}) \\ = J_{\infty} (u_{n}) + \frac{1}{p} \int_{R^{N}} (V (ϵ x) - V_{\infty}) | u_{n} |^{p} d x \\ = J_{\infty} (u_{n}) + \frac{1}{p} \int_{B_{R / ϵ} (0)} (V (ϵ x) - V_{\infty}) | u_{n} |^{p} d x + \frac{1}{p} \int_{R^{N} ∖ B_{R / ϵ} (0)} (V (ϵ x) - V_{\infty}) | u_{n} |^{p} d x \\ \geq J_{\infty} (u_{n}) + \frac{1}{p} \int_{B_{R / ϵ} (0)} (V (ϵ x) - V_{\infty}) | u_{n} |^{p} d x - \frac{1}{p} ζ \int_{R^{N} ∖ B_{R / ϵ} (0)} | u_{n} |^{p} d x \\ \geq J_{\infty} (u_{n}) - C_{3} ζ + o_{n} (1) \geq E_{\infty, a} - C_{3} ζ + o_{n} (1) . \end{aligned}

Since

ζ > 0

is arbitrary, this implies that

E_{0, a} + ρ_{1} \geq E_{\infty, a}

, which contradicts with the definition of

ρ_{1}

. Thus, we finish the proof of the lemma.

Lemma 3.3

Let ${u_{n}} \subset S_{a}$ be a $(P S)_{c}$ sequence for $J_{ϵ}$ restricted to $S_{a}$ with $c < E_{0, a} + ρ_{1} < 0$ and $u_{n} ⇀ u_{ϵ}$ in $W^{1, p} (R^{N})$ . If $v_{n} := u_{n} - u_{ϵ} ↛ 0$ in $W^{1, p} (R^{N})$ , then decreasing $ϵ_{0}$ if necessary, there is $β > 0$ independent of $ϵ \in (0, ϵ_{0})$ such that

\begin{aligned} \underset{n \to \infty}{lim inf} ‖ u_{n} - u_{ϵ} ‖_{p}^{p} \geq β . \end{aligned}

Proof.

Define

\begin{aligned} Ψ (u) = \frac{1}{p} \int_{R^{N}} | u |^{p} d x . \end{aligned}

then

S_{a} = Ψ^{- 1} ({a^{p} / p})

. Applying Proposition 5.12 in Willem (1996), there exists a sequence

{λ_{n}} \subset R

such that

\begin{aligned} ‖ J_{ϵ}^{'} (u_{n}) - λ_{n} Ψ^{'} (u_{n}) ‖_{(W^{1, p} (R^{N}))^{'}} \to 0 as n \to \infty . \end{aligned}

(3.2)

By Lemma 2.1,

{u_{n}}

is bounded in

W^{1, p} (R^{N})

. Therefore, we can see that

λ_{n}

is bounded. We can assume that

λ_{n} \to λ_{ϵ}

n \to \infty

. This yields

\begin{aligned} ‖ J_{ϵ}^{'} (u_{n}) - λ_{ϵ} Ψ^{'} (u_{n}) ‖_{(W^{1, p} (R^{N}))^{'}} \to 0 as n \to \infty \end{aligned}

and

\begin{aligned} J_{ϵ}^{'} (u_{ϵ}) - λ_{ϵ} Ψ^{'} (u_{ϵ}) = 0 in (W^{1, p} (R^{N}))^{'} . \end{aligned}

Using the method from Zhu et al. (2025), we can prove that

\begin{aligned} ‖ J_{ϵ}^{'} (v_{n}) - λ_{ϵ} Ψ^{'} (v_{n}) ‖_{(W^{1, p} (R^{N}))^{'}} \to 0 as n \to \infty . \end{aligned}

By (3.2), we have

\begin{aligned} 0 & > E_{0, a} + ρ_{1} > c \\ = \underset{n \to \infty}{lim inf} J_{ϵ} (u_{n}) \\ = \underset{n \to \infty}{lim inf} [J_{ϵ} (u_{n}) - \frac{1}{p} J_{ϵ}^{'} (u_{n}) u_{n} + \frac{λ_{n}}{p} a^{p}] \\ = \underset{n \to \infty}{lim inf} [(\frac{1}{p} - \frac{1}{2 q}) \int_{R^{N}} (I_{α} * | u_{n} |^{q}) | u_{n} |^{q} d x + (\frac{μ}{p} - \frac{μ}{s}) \int_{R^{N}} | u_{n} |^{s} d x + \frac{λ_{n}}{p} a^{p}] \\ \geq \frac{λ_{ϵ}}{p} a^{p} . \end{aligned}

As a result,

\begin{aligned} \underset{ϵ \to 0^{+}}{lim sup} λ_{ϵ} \leq \frac{p (ρ_{1} + E_{0, c})}{a^{p}} < 0. \end{aligned}

Therefore, there exists a constant

λ_{*}

independent of

ϵ

such that

\begin{aligned} λ_{ϵ} \leq λ_{*} < 0 \forall ϵ \in (0, ϵ_{0}) . \end{aligned}

In addition, since ${v_{n}}$ is bounded, we get

\begin{aligned} \int_{R^{N}} | \nabla v_{n} |^{p} + V (ϵ x) | v_{n} |^{p} d x - λ_{ϵ} \int_{R^{N}} | v_{n} |^{p} d x = \int_{R^{N}} (I_{α} * | v_{n} |^{q}) | v_{n} |^{q} d x + μ \int_{R^{N}} | v_{n} |^{s} d x + o_{n} (1) . \end{aligned}

It follows that

\begin{aligned} \int_{R^{N}} | \nabla v_{n} |^{p} + V (ϵ x) | v_{n} |^{p} d x - λ_{*} \int_{R^{N}} | v_{n} |^{p} d x \leq \int_{R^{N}} (I_{α} * | v_{n} |^{q}) | v_{n} |^{q} d x + μ \int_{R^{N}} | v_{n} |^{s} d x + o_{n} (1) . \end{aligned}

Combining Lemma 1.1 and

(V)

, we deduce that

\begin{aligned} ‖ v_{n} ‖_{\frac{2 N q}{N + β}}^{2 q} + μ ‖ v_{n} ‖_{s}^{s} + o_{n} (1) \geq ‖ v_{n} ‖ . \end{aligned}

Since

v_{n} ↛ 0

W^{1, p} (R^{N})

, we may assume that

\underset{n \to \infty}{lim inf} ‖ v_{n} ‖ > 0

. Applying the continuous Sobolev embedding

W^{1, p} (R^{N}) ↪ L^{r} (R^{N})

, we obtain

\begin{aligned} \underset{n \to \infty}{lim inf} ‖ v ‖ \geq C_{4} . \end{aligned}

Then, there holds

\begin{aligned} C_{5} & \leq \underset{n \to \infty}{lim inf} ‖ v_{n} ‖_{s}^{s} \\ \leq \underset{n \to \infty}{lim inf} [C_{N, s} ‖ \nabla v_{n} ‖_{p}^{s δ_{s}} ‖ v_{n} ‖_{p}^{s (1 - δ_{s})}] \\ \leq C_{6} \underset{n \to \infty}{lim inf} ‖ v_{n} ‖_{p}^{s (1 - δ_{s})} . \end{aligned}

Furthermore, we can derive

\begin{aligned} \underset{n \to \infty}{lim inf} ‖ v_{n} ‖_{p}^{p} \geq {(\frac{C_{5}}{C_{6}})}^{\frac{p}{s (1 - δ_{s})}} > 0. \end{aligned}

Hereafter, in order to prove that $J_{ϵ}$ satisfies the $(P S)_{c}$ condition, we fix

\begin{aligned} 0 < ρ < min {\frac{1}{2}, \frac{β}{a^{p}}} (E_{\infty, a} - E_{V_{0}, a}) \leq ρ_{1} . \end{aligned}

Lemma 3.4

For each $ϵ \in (0, ϵ_{0})$ , the functional $J_{ϵ}$ satisfies the $(P S)_{c}$ condition restricted to $S_{a}$ for $c < E_{0, a} + ρ$ .

Proof.

Let ${u_{n}}$ be a $(P S)_{c}$ sequence for $J_{ϵ}$ restricted to $S_{a}$ . Set

\begin{aligned} Ψ (u) = \frac{1}{p} \int_{R^{N}} | u |^{p} d x . \end{aligned}

Thus,

Ψ^{- 1} ({a^{p} / p})

. From Proposition 5.12 in Willem (1996), there exists a sequence

{λ_{n}} \subset R

such that

\begin{aligned} ‖ J_{ϵ}^{'} (u_{n}) - λ_{n} Ψ^{'} (u_{n}) ‖_{(W^{1, p} (R^{N}))^{'}} \to 0 as n \to \infty . \end{aligned}

By Lemma 3.3, if

v_{n} = u_{n} - u_{ϵ} ↛ 0

W^{1, p} (R^{N})

, then there exists

β > 0

such that

\begin{aligned} \underset{n \to \infty}{lim inf} ‖ v_{n} ‖_{p}^{p} \geq β > 0. \end{aligned}

(3.3)

Let $d_{n} = ‖ v_{n} ‖_{p}$ and $b = ‖ u_{ϵ} ‖_{p}$ . Suppose that $d_{n} \to d$ , then (3.3) gives $d^{p} \geq β > 0$ , and Lemma 3.2 ensures $b > 0$ . Applying the Brézis–Lieb lemma we can see that

\begin{aligned} ‖ u ‖_{p}^{p} = ‖ v_{n} ‖_{p}^{p} + ‖ u_{ϵ} ‖_{p}^{p} + o_{n} (1) . \end{aligned}

Then

a^{p} = b^{p} + d^{p}

, which implies that

a > d_{n}

for large

n

. Employing the same technical as in the proof of Lemma 3.2, we have

\begin{aligned} J_{ϵ} (v_{n}) \geq E_{\infty, d_{n}} + o_{n} (1) . \end{aligned}

As a result, using condition

(V)

together with Lemma 2.3, we conclude that

\begin{aligned} E_{0, a} + ρ + o_{n} (1) & \geq c + o_{n} (1) = J_{ϵ} (u_{n}) \\ = J_{ϵ} (v_{n}) + J_{ϵ} (u_{ϵ}) + o_{n} (1) \\ \geq E_{\infty, d_{n}} + E_{0, b} + o_{n} (1) \\ \geq \frac{d_{n}^{p}}{a^{p}} E_{\infty, a} + \frac{b^{p}}{a^{p}} E_{0, a} + o_{n} (1) . \end{aligned}

Taking the limit as

n \to \infty

, we obtain

\begin{aligned} ρ \geq \frac{d^{p}}{a^{p}} (E_{\infty, a} - E_{0, a}) \geq \frac{β}{a^{p}} (E_{\infty, a} - E_{0, a}), \end{aligned}

obviously, we can get a contradiction since

ρ < β / a^{p} (E_{\infty, a} - E_{0, a})

, and finishing the proof.

By Lemma 3.4, we have $v_{n} \to 0$ in $W^{1, p} (R^{N})$ , namely, $u_{n} \to u_{ϵ}$ in $W^{1, p} (R^{N})$ . Consequently, $‖ u_{ϵ} ‖_{p} = a$ and

\begin{aligned} - Δ_{p} u_{ϵ} + V (ϵ x) | u_{ϵ} |^{p - 2} u_{ϵ} = λ_{ϵ} | u_{ϵ} |^{p - 2} u_{ϵ} + (I_{α} * | u_{ϵ} |^{q}) | u_{ϵ} |^{q - 2} u_{ϵ} + μ | u_{ϵ} |^{s - 2} u_{ϵ} . \end{aligned}

Summarizing, from the above discussions, we can find that problem (1.1) has at least a normalized solutions, and the existence result is obtained.

4. The Multiplicity Result

In this section, we investigate the multiplicity of normalized solutions for problem (1.1) by the Lusternik–Schnirelmann category theory and study the behavior of its maximum points concentrating on the set $M$ of global minima of the potential $V$ . In order to achieve our aim, we recall the following result for critical points involving Lusternik–Schnirelmann category. We refer to Benci and Cerami (1994) and Cingolani and Lazzo (2000) for more details.

Proposition 4.1

Let $U$ be a complete $C^{1}$ Riemannian manifold. Assume that $φ \in C^{1} (U, R)$ is bounded from below and satisfies $- \infty < inf_{U} φ < d < k < \infty$ . Moreover, suppose that $φ$ satisfies the Palais–Smale condition on the sublevel ${u \in U : φ (u) \leq k}$ and that $d$ is not a critical level for $φ$ . Then

\begin{aligned} c a r d {u \in φ^{d} : φ^{'} (u) = 0} \geq c a t_{φ^{d}} (φ^{d}), \end{aligned}

where

φ^{d} = {u \in U : φ (u) \leq d}

With a view to employ Proposition 4.1, the following abstract lemma provides a very useful tool since relates the topology of some sublevel of a functional to the topology of some subset of the space $R^{N}$ , the main proof can be found in Benci and Cerami (1994).

Proposition 4.2

Let $Ω$ , $Ω_{1}$ and $Ω_{2}$ be closed sets with $Ω_{1} \subset Ω_{2}$ and let $π : Ω \to Ω_{2}$ , $ψ : Ω_{1} \to Ω$ be continuous maps such that $π \circ ψ$ is homotopically equivalent to the inclusion mapping $j : Ω_{1} \to Ω_{2}$ . Then $c a t_{Ω} (Ω) \geq c a t_{Ω_{2}} (Ω_{1})$ .

Next, based on the constructive techniques, we will introduce two maps $Φ_{ϵ}$ and $β_{ϵ}$ such that their composition $β_{ϵ} \circ Φ_{ϵ}$ is homotopically equivalent to the inclusion mapping $j : M \to M_{δ}$ .

From Lemma 2.5 and Remark 2.1, we can find that there exists $(w, λ) \in W^{1, p} (R^{N}) \times R$ such that it solve the following problem

\begin{aligned} {\begin{cases} - Δ_{p} u + V_{0} | u |^{p - 2} u = λ | u |^{p - 2} u + (I_{α} * | u |^{q}) | u |^{q - 2} u + μ | u |^{s - 2} u in R^{N}, \\ \int_{R^{N}} | u |^{p} d x = a^{p} > 0, \end{cases} \end{aligned}

(4.1)

and

J_{0} (w) = E_{0, a}

. Let

η

be a smooth cut-off function defined in

[0, \infty)

such that

η (s) = 1

s \leq 1 / 2

and

η (s) = 0

s \geq 1

. For each

y \in M

, we define

\begin{aligned} Ψ_{ϵ, y} (x) := η (| ϵ x - y |) w (\frac{ϵ x - y}{ϵ}), \\ {\tilde{Ψ}}_{ϵ, y} (x) := a \frac{Ψ_{ϵ, y} (x)}{‖ Ψ_{ϵ, y} ‖_{p}}, \end{aligned}

and

Φ_{ϵ} : M \to S_{a}

Φ_{ϵ} (y) = {\tilde{Ψ}}_{ϵ, y}

. According to the construction of

Ψ_{ϵ, y}

, we can see that

Φ_{ϵ} (y)

has compact support for any

y \in M

. Moreover, we have the following conclusion for

Φ_{ϵ}

Lemma 4.1

The function $Φ_{ϵ}$ satisfies

\begin{aligned} lim_{ϵ \to 0} J_{ϵ} (Φ_{ϵ} (y)) = E_{0, a} uniformly in y \in M . \end{aligned}

Proof.

Arguing by contradiction, we can assume that there exist $ϵ_{0} > 0$ , ${y_{n}} \subset M$ , and $ϵ_{n} \to 0$ such that

\begin{aligned} | J_{ϵ_{n}} (Φ_{ϵ_{n}} (y_{n})) - E_{0, a} | \geq ϵ_{0} . \end{aligned}

By Lebesgue dominated convergence theorem, we deduce that

\begin{aligned} lim_{n \to \infty} \int_{R^{N}} | \nabla Φ_{ϵ_{n}} (y_{n}) |^{p} d x = lim_{n \to \infty} \int_{R^{N}} \frac{a^{p}}{‖ Ψ_{ϵ_{n}, y_{n}} ‖_{p}^{p}} | \nabla η (ϵ_{n} z) w (z) |^{p} d x = \int_{R^{N}} | \nabla w |^{p} d x, \\ lim_{n \to \infty} \int_{R^{N}} V (ϵ_{n} x) | Φ_{ϵ_{n}} (y_{n}) |^{p} d x = lim_{n \to \infty} \int_{R^{N}} \frac{a^{p} V (ϵ_{n} z + y_{n})}{‖ Ψ_{ϵ_{n}, y_{n}} ‖_{p}^{p}} | η (ϵ_{n} z) w (z) |^{p} d x = \int_{R^{N}} V_{0} | w |^{p} d x, \\ lim_{n \to \infty} \int_{R^{N}} (I_{α} * {| a \frac{Ψ_{ϵ_{n}, y_{n}} (x)}{‖ Ψ_{ϵ_{n}, y_{n}} ‖_{p}} |}^{q}) {| a \frac{Ψ_{ϵ_{n}, y_{n}} (x)}{‖ Ψ_{ϵ_{n}, y_{n}} ‖_{p}} |}^{q} d x = \int_{R^{N}} (I_{α} * | w |^{q}) | w |^{q} d x, \end{aligned}

and

\begin{aligned} lim_{n \to \infty} \int_{R^{N}} | Φ_{ϵ_{n}} (y_{n}) |^{s} d x = \int_{R^{N}} | w |^{s} d x . \end{aligned}

Consequently, we have

\begin{aligned} J_{ϵ_{n}} (Φ_{ϵ_{n}} (y_{n})) & = \frac{1}{p} \int_{R^{N}} | \nabla Φ_{ϵ_{n}} (y_{n}) |^{p} d x + \frac{1}{p} \int_{R^{N}} V (ϵ_{n} x) | Φ_{ϵ_{n}} (y_{n}) |^{p} d x \\ + \frac{1}{2 q} \int_{R^{N}} (I_{α} * | Φ_{ϵ_{n}} (y_{n}) |^{q}) | Φ_{ϵ_{n}} (y_{n}) |^{q} d x + + \frac{μ}{s} \int_{R^{N}} | Φ_{ϵ_{n}} (y_{n}) |^{s} d x \\ \to \frac{1}{p} \int_{R^{N}} | \nabla w |^{p} d x + \frac{1}{p} \int_{R^{N}} V_{0} | w |^{p} d x + \frac{1}{2 q} \int_{R^{N}} (I_{α} * | w |^{q}) | w |^{q} d x + \frac{μ}{s} \int_{R^{N}} | w (x) |^{s} d x \\ = J_{0} (w) = E_{0, a}, \end{aligned}

which is impossible. The proof is completed.

Now, we introduce the barycenter map. For any $δ > 0$ , let $ρ = ρ (δ) > 0$ be such that $M_{δ} \subset B_{ρ} (0)$ . We define $χ : R^{N} \to R^{N}$ as follows

\begin{aligned} χ (x) = {\begin{cases} x & if | x | \leq ρ, \\ \frac{ρ x}{| x |} & if | x | \geq ρ . \end{cases} \end{aligned}

The barycenter map

β_{ϵ} (u) : S_{a} \to R^{N}

is defined by

\begin{aligned} β_{ϵ} (u) = \frac{\int_{R^{N}} χ (ϵ x) | u |^{p} d x}{\int_{R^{N}} | u |^{p} d x} . \end{aligned}

Lemma 4.2

The function $β_{ϵ}$ satisfies

\begin{aligned} lim_{ϵ \to 0} β_{ϵ} (Φ (y)) = y uniformly in y \in M . \end{aligned}

Proof.

We argue by contradiction. So, we suppose that there exist $δ_{0} > 0$ , ${y_{n}} \subset M$ , and $ϵ_{n} \to 0$ such that

\begin{aligned} | β_{ϵ_{n}} (Φ_{ϵ_{n}} (y_{n})) - y_{n} | \geq δ_{0}, \forall n \in N . \end{aligned}

(4.2)

According to the definition of

Φ_{ϵ_{n}}

and

β_{ϵ_{n}}

, we can infer that

\begin{aligned} β_{ϵ_{n}} (Φ_{ϵ_{n}} (y_{n})) = y_{n} + \frac{\int_{R^{N}} [χ (ϵ_{n} x + y_{n}) - y_{n}] η^{p} (| ϵ_{n} x |) w^{p} (x) d x}{\int_{R^{N}} η^{p} (| ϵ_{n} x |) w^{p} (x) d x} . \end{aligned}

Since

{y_{n}} \subset M \subset B_{ρ} (0)

, employing the Lebesgue dominating convergence theorem, we have

\begin{aligned} | β_{ϵ_{n}} (Φ (y_{n})) - y_{n} | \to 0 as n \to \infty, \end{aligned}

which contradicts relation (4.2).

Lemma 4.3

Let $ϵ_{n} \to 0$ and ${u_{n}} \subset S_{a}$ with $J_{ϵ_{n}} (u_{n}) \to E_{0, a}$ . Then, there exists a sequence ${y_{n}} \subset R^{N}$ such that: (i)

$v_{n} (x) = u_{n} (x + y_{n}) \to v \neq 0$ in $W^{1, p} (R^{N})$ ,

(ii)

$ϵ_{n} y_{n} \to y \in M$ .

Proof.

From Lemma 2.1, it follows that ${u_{n}}$ is bounded in $W^{1, p} (R^{N})$ . We claim that there exist $R$ , $γ > 0$ and $y_{n} \subset R^{N}$ such that

\begin{aligned} \int_{B_{R} (y_{n})} | u_{n} |^{p} d x \geq γ \forall n \in N . \end{aligned}

Otherwise, we have

u_{n} \to 0

L^{r} (R^{N})

for all

r \in (p, p^{*})

. Similar to the proof of Lemma 2.4, we can get

\begin{aligned} E_{0, a} = lim_{n \to \infty} J_{ϵ_{n}} (u_{n}) \geq 0, \end{aligned}

which contradicts the fact that

E_{0, a} < 0

. Set

v_{n} := u_{n} (x + y_{n})

. Passing to a subsequence, we may assume that

v_{n} ⇀ v \neq 0

. Observe that

\begin{aligned} E_{0, a} \leq J_{0} (v_{n}) = J_{0} (u_{n}) \to E_{0, a} as n \to \infty . \end{aligned}

From Lemma 2.4, we obtain

v_{n} \to v

W^{1, p} (R^{N})

and

v \in S_{a}

Set ${\tilde{y}}_{n} := ϵ_{n} y_{n}$ , we prove ${\tilde{y}}_{n} := ϵ_{n} y_{n}$ is bounded in $R^{N}$ . In fact, if $| y_{n} | \to \infty$ , then

\begin{aligned} E_{0, a} & = lim_{n \to \infty} J_{ϵ_{n}} (u_{n}) \\ = lim_{n \to \infty} {\frac{1}{p} \int_{R^{N}} | \nabla u_{n} |^{p} d x + \frac{1}{p} \int_{R^{N}} V (ϵ_{n} x) | u_{n} |^{p} d x - \frac{1}{2 q} \int_{R^{N}} (I_{α} * | u_{n} |^{q}) | u_{n} |^{q} d x - \frac{μ}{s} \int_{R^{N}} | u_{n} |^{s} d x} . \\ = lim_{n \to \infty} {\frac{1}{p} \int_{R^{N}} | \nabla v_{n} |^{p} d x + \frac{1}{p} \int_{R^{N}} V (ϵ_{n} x + {\tilde{y}}_{n}) | v_{n} |^{p} d x - \frac{1}{2 q} \int_{R^{N}} (I_{α} * | v_{n} |^{q}) | v_{n} |^{q} d x - \frac{μ}{s} \int_{R^{N}} | v_{n} |^{s} d x} . \\ = J_{\infty} (v) \geq E_{\infty, a}, \end{aligned}

which contradicts relation (3.1). Therefore, there exists a subsequence, still denoted by

{\tilde{y}}_{n}

such that

{\tilde{y}}_{n} \to y

. With a similar arguments as the above inequality we obtain

\begin{aligned} E_{0, a} & \geq lim_{n \to \infty} [\frac{1}{p} \int_{R^{N}} | \nabla v |^{p} + V (y) | v |^{p} d x - \frac{1}{2 q} \int_{R^{N}} (I_{α} * | v |^{q}) | v |^{q} d x - \frac{μ}{s} \int_{R^{N}} | v |^{s} d x] \\ = J_{V (y)} (v) \geq E_{V (y), a} . \end{aligned}

Noting that

V (y) \geq V_{0}

. If

V (y) > V_{0}

, using Corollary 2.1, we can get a contradiction. As a consequence,

V (y) = V_{0}

, and

y \in M

Let $ℓ : [0, + \infty) \to [0, + \infty)$ be positive function given by

\begin{aligned} ℓ (ϵ) := max_{y \in M} | J_{ϵ} (Φ_{ϵ} (y)) - E_{0, a} | . \end{aligned}

From Lemma 4.1 we can deduce that the function

ℓ (ϵ)

satisfies

ℓ (ϵ) \to 0

ϵ \to 0

. Define a subset

{\tilde{S}}_{a}

S_{a}

, and set

\begin{aligned} {\tilde{S}}_{a} = {u \in S_{a} : J_{ϵ} (u) \leq E_{0, a} + ℓ (ϵ)} . \end{aligned}

Then,

Φ_{ϵ} (y) \in {\tilde{S}}_{a}

for all

y \in M

, and

{\tilde{S}}_{a} \neq \emptyset

. Moreover, we have the following result.

Lemma 4.4

For any $δ > 0$ , then the following conclusion holds

\begin{aligned} lim_{ϵ \to 0} sup_{u \in \tilde{S} (a)} inf_{y \in M_{δ}} | β_{ϵ} (u) - y | = 0. \end{aligned}

Proof.

Let $ϵ_{n} \to 0$ and $u_{n} \in {\tilde{S}}_{a}$ be such that

\begin{aligned} inf_{y \in M_{δ}} | β_{ϵ_{n}} (u_{n}) - y | = sup_{u \in \tilde{S} (a)} inf_{y \in M_{δ}} | β_{ϵ_{n}} (u_{n}) - y | + o_{n} (1) . \end{aligned}

Then, it is sufficient to prove that there exists

{y_{n}} \subset M_{δ}

such that

\begin{aligned} lim_{n \to \infty} | β_{ϵ_{n}} (u_{n}) - y_{n} | = 0. \end{aligned}

Since

u_{n} \in \tilde{S} (a)

\begin{aligned} E_{0, a} \leq J_{0} (u_{n}) \leq J_{ϵ_{n}} (u_{n}) \leq E_{0, a} + ℓ (ϵ_{n}) \forall n \in N, \end{aligned}

we can see that

u_{n} \in S_{a}

and

J_{ϵ_{n}} (u_{n}) \to E_{0, a}

. From Lemma 4.3, there exists

{y_{n}} \subset R^{N}

such that

{\tilde{y}}_{n} := ϵ_{n} y_{n} \to y

for some

y \in M

and

v_{n} (x) := u_{n} (x + y_{n}) \to v

W^{1, p} (R^{N})

with

v \neq 0

. Therefore, we can deduce that

{{\tilde{y}}_{n}} \subset M_{δ}

for large

n

and

\begin{aligned} β_{ϵ_{n}} (u_{n}) - {\tilde{y}}_{n} = \frac{\int_{R^{N}} [χ (ϵ_{n} z + {\tilde{y}}_{n}) - {\tilde{y}}_{n}] | v_{n} |^{p} d z}{a^{p}} \to 0 as n \to \infty . \end{aligned}

We finish the proof of the Lemma.

Now we are in a position to complete the proofs of Theorem 1.1.

Proof of Theorem 1.1: We will take two parts to complete the proof.

(i) Multiplicity of solutions. For any $y \in M$ , from Lemmas 4.1 and 4.4, we can deduce that there exists $ϵ_{δ} > 0$ such that for every $ϵ \in (0, ϵ_{δ})$ , the diagram

\begin{aligned} β_{ϵ} \circ Φ_{ϵ} : M \to {\tilde{S}}_{a} \to M_{δ} \end{aligned}

is well defined. By Lemma 4.2, there exists a function

κ (ϵ, y)

with

| κ (ϵ, y) | < (δ / 2)

uniformly in

y \in M

for all

ϵ \in (0, ϵ_{δ})

, such that

β_{ϵ} (Φ_{ϵ} (y)) = y + κ (ϵ, y)

for all

y \in M

. We define the function

\begin{aligned} H (t, y) = y + (1 - t) κ (ϵ, y) . \end{aligned}

Then,

H : [0, 1] \times M \to M_{δ}

is continuous. Evidently,

H (0, y) = β_{ϵ} (Φ_{ϵ} (y))

and

H (1, y) = y

for all

y \in M

, and

β_{ϵ} \circ Φ_{ϵ}

is homotopic to the inclusion mapping

j : M \to M_{δ}

. Applying Proposition 4.2 we can obtain

\begin{aligned} c a t (\tilde{S} (a)) \geq c a t_{M_{δ}} (M) . \end{aligned}

On the other hand, let us choose a function $ℓ (ϵ) > 0$ such that $ℓ (ϵ) \to 0$ as $ϵ \to 0$ and such that $E_{0, a} + ℓ (ϵ)$ is not a critical level for $J_{ϵ}$ . Furthermore, by Lemma 3.4 we see that $J_{ϵ}$ satisfies the $(P S)_{c}$ condition at levels $c \in (E_{0, a}, E_{0, a} + ℓ (ϵ))$ Therefore, by Proposition 4.2, we can conclude that $J_{ϵ}$ has at least $c a t_{M_{δ}} (M)$ critical point on $S_{a}$ .

(ii) Concentration of solutions. For any $ϵ_{n} \to 0$ , let $(u_{ϵ_{n}}, λ_{n})$ be a solution to problem (1.1) and $ξ_{n}$ be a global maximum of $| u_{ϵ_{n}} |$ . Thus,

\begin{aligned} E_{0, a} \leq J_{0} (u_{ϵ_{n}}) \leq J_{ϵ_{n}} (u_{ϵ_{n}}) \leq E_{0, a} + ℓ (ϵ_{n}), \end{aligned}

which implies that

J_{ϵ_{n}} (u_{ϵ_{n}}) \to E_{0, a}

n \to \infty

. From Lemma 4.3, there exists

{y_{n}} \in R^{N}

with

{\tilde{y}}_{n} := ϵ_{n} y_{n} \to y \in M

such that

v_{n} (x) := u_{ϵ_{n}} (x + y_{n}) \to v \in W^{1, p} (R^{N}) ∖ {0}

. Then we can see that

\begin{aligned} - Δ_{p} v_{n} + V (ϵ_{n} x + {\tilde{y}}_{n}) | v_{n} |^{p - 2} v_{n} = λ | v_{n} |^{p - 2} v_{n} + (I_{α} * | v_{n} |^{q}) | v_{n} |^{q - 2} v_{n} + μ | v_{n} |^{s - 2} v_{n} . \end{aligned}

Similar to the proof of Lemma 3.3, we obtain

\begin{aligned} \underset{ϵ \to 0}{lim sup} λ_{n} \leq \frac{ρ_{1} + E_{0, a}}{a^{p}} < 0. \end{aligned}

Since

v_{n} \to v

W^{1, p} (R^{N})

, then we have

lim_{| x | \to \infty} v_{n} (x) = 0

uniformly for all

n \in N

. Then, for any fix

τ > 0

, there are

R_{1} > 0

and

n_{0} \in N

such that

| v_{n} (x) | \leq τ

for

| x | \geq R_{1}

and

n \geq n_{0}

. We claim that

‖ v_{n} ‖_{\infty} ↛ 0

n \to \infty

. Otherwise,

v_{n} \to 0

W^{1, p} (R^{N})

, which contradicts to

v_{n} \in S_{a}

. Let us fix

τ > 0

such that

‖ v_{n} ‖_{\infty} \geq 2 τ

and

z_{n} \in R^{N}

satisfying

| v_{n} (z_{n}) | = ‖ v_{n} ‖_{\infty}

for all

n \in N

. Then,

ξ_{n} = z_{n} + y_{n}

and

\begin{aligned} lim_{n \to + \infty} V (ϵ_{n} ξ_{n}) = lim_{n \to + \infty} V (ϵ_{n} z_{n} + ϵ_{n} y_{n}) = V (y) = V_{0} . \end{aligned}

We finish the proof of Theorem 1.1.□

Footnotes

Acknowledgments

This work was supported by the National Natural Science Foundation of China (12271152), and the Key project of Scientific Research Project of Department of Education of Hunan Province (24A0430, 23A0478).

Funding

The authors received no financial support for the research, authorship and/or publication of this article.

Declaration of Conflicting Interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data Availability Statements

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

ORCID iD

Jian Zhang

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