The existence of multiple solutions for a class of upper critical Choquard equation in a bounded domain

1 Introduction and main results

The purpose of this article is to investigate the existence of multiple solutions to the following critical Choquard equation:

where μ > 0 is a parameter, N > 4 , 0 < α < N , 2 α * = N + α N − 2 , I α is the Riesz potential, defined for every x ∈ R N \ { 0 } by

where Γ is the Gamma function, N N − 2 < p < 2 , Ω ⊂ R N is a bounded domain with smooth boundary, and f and g are continuous functions. Problem (1.1) is related to the stationary solutions of equation:

which was proposed by Pekar in 1954 to model the quantum theory of a polaron at rest and considered by Choquard as a certain approximation to the Hartree-Fock theory of one-component plasma and by Penrose for self-gravitating matter. For more information about the physical background, one can refer to [1–3] and references therein.

Owing to the appearance of the terms I α * ( g ∣ u ∣ 2 α * ) , Problem (1.1) is nonlocal. This leads to some mathematical difficulties and makes studying such problems more interesting. After the pioneering work of [4,5], it has received much attention. The existence and qualitative properties of solutions for the Choquard equation have been studied a lot (see [6–10] for the existence of normalized solutions, see [11–15] for the existence of ground-state solutions, and for more results about the Choquard equation, we refer to [16–26]).

Alves et al. [27] considered the following critical Choquard equation:

where 0 < μ < 3 , ε is the positive parameter and V and Q are two continuous real functions on R 3 . Using the category theory, they established the existence of multiple solutions. For ε small, they also obtained the concentration phenomenon.

Meng and He [28] studied the following upper critical Choquard equation:

where N ≥ 3 , ( N − 4 ) + < α < N , ε is the small parameter, V ( x ) ∈ C ( R N ) ∩ L ∞ ( R N ) is the positive potential and f ∈ C 1 ( R + , R ) is the subcritical nonlinear term. Using Nehari manifold and delicate compactness analyses, they obtained k different positive solutions. When ε > 0 is small, the concentration phenomenon of these solutions is also obtained.

Su and Liu [29] considered the nonlinear Choquard equation with critical growth:

where N ⩾ 4 , α ∈ ( 0 , N ) , and F ( u ) ≔ 1 q ∣ u ∣ q + 1 2 α * ∣ u ∣ 2 α * . When the parameter ε is small enough, they obtained a positive single-peak solution for each q ∈ ( 2 α ♯ , 2 α * ) . This result extends some results established in Cingolani and Tanaka [30] for the case q < 2 , which was seen as an open problem in Moroz and Van Schaftingen [31].

Li et al. [32] considered the critical Choquard equation with concave perturbation:

where α ∈ ( 0 , 3 ) , f = F ′ ∈ C ( R , R ) , and g ∈ C ( R 3 × R , R ) . Whether for the upper critical, lower critical, or double critical cases, they can obtain two positive solutions when the parameter is small enough, one of which is a ground-state solution.

There are few results about the existence of multiple solutions for the Choquard equation in a bounded domain. Lin [33] studied the existence and multiplicity of positive solutions for the Dirichlet problem:

Under the conditions N N − 2 < q < 2 and N > 4 , they proved that there exist k + 1 positive solutions for the aforementioned problem. Liao et al. [34] complemented and optimized their work and obtained the same results for the aforementioned problem with 1 < q < 2 and N ≥ 3 .

Motivated by aforementioned works, this article is devoted to study the existence of multiple solutions for Problem (1.1). We will study how the shape of the potential g ( x ) affects the number of the solutions of (1.1). To study Problem (1.1) variationally, we need the following celebrated results.

To state our main results, we need some assumptions about the potentials f and g .

1. f , g ∈ C ( Ω ¯ ) and f ( x ) > 0 , g ( x ) > 0 , for x ∈ Ω ¯ .

2. There exist k points a 1 , a 2 , … , a k in Ω such that g ( a i ) are strict local maxima satisfying

   g ( a i ) = max x ∈ Ω ¯ g ( x ) = 1 ,

   and there exists a positive number β ∈ [ N − 2 2 , N + α 2 ) such that g ( x ) − g ( a i ) = O ( ∣ x − a i ∣ β ) as x → a i for i = 1 , … , k .

Now, we state our main results.

This article is organized as follows. In Section 2, the variational setting is set up and some preliminary results are given. In Section 3, some properties of a barycenter map are studied. In Section 4, we give the proof of Theorem 1.6.

Notation. In this article, we make use of the following notations:

• B R ( x ) denotes the open ball of radius R centered at x , where R > 0 and x ∈ R N .

• The letter C stands for any positive constants.

• “ → ” and “ ⇀ ” represent strong convergence and weak convergence, respectively.

• o n ( 1 ) is a quantity tending to 0 as n → ∞ .

• ∣ u ∣ s = ( ∫ R N ∣ u ∣ s d x ) 1 s denotes the norm of u in L s ( R N ) for s ≥ 1 .

• ∣ u ∣ ∞ = sup x ∈ Ω ∣ u ( x ) ∣ denotes the norm of u in L ∞ ( Ω ) .

• ∣ Ω ∣ is the Lebesgue measure of Ω ⊂ R N .

• H 0 1 ( Ω ) is the closure of C 0 ∞ ( Ω ) in H 1 ( R N ) , where H 1 ( R N ) ≔ { u ∈ L 2 ( R N ) : ∇ u ∈ L 2 ( R N ) } with the norm ∫ R N ( ∣ ∇ u ∣ 2 + u 2 ) 1 2 .

• ‖ u ‖ = ∫ Ω ∣ ∇ u ∣ 2 d x 1 ∕ 2 denotes the norm of u in H 0 1 ( Ω ) .

• D 1 , 2 ( R N ) ≔ { u ∈ L 2 * ( R N ) : ∇ u ∈ L 2 ( R N ) } with the norm ∫ R N ∣ ∇ u ∣ 2 1 2 .

2 Preliminaries

For Problem (1.1), the associated energy functional is

where

It is easy to check that I μ is well defined on H 0 1 ( Ω ) and I μ ∈ C 1 ( H 0 1 ( Ω ) , R ) . Define the Nehari manifold

Let

By direct computation, we have

Then, for u ∈ N μ , one has

and

We can divide N μ into three parts:

By Lemma 2.1,

are well defined.