Existence of ground state solutions for critical fractional Choquard equations involving periodic magnetic field

1 Introduction

This paper is concerned with the following fractional Choquard equation involving magnetic field and critical nonlinearity:

where ε is a small positive parameter, s ∈ ( 0 , 1 ) , α ∈ ( 0 , N ) , N > max { 2 μ + 4 s , 2 s + α ∕ 2 } , and 2 s , α ∗ = 2 N − α N − 2 s is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality. The fractional magnetic Laplacian ( − Δ ) A s has been introduced in [16,24] with motivations falling into the framework of the general theory of Lévy processes, and up to normalization constants, is defined for any u ∈ C c ∞ ( R N , C ) as follows:

where B r ( x ) denotes the ball in R N of center at x and radius r > 0 , which can be considered as the fractional counterpart of the classical magnetic Laplacian 1 i ∇ − A 2 . If A ≡ 0 , the operator ( − Δ ) A s becomes the celebrated fractional Laplacian ( − Δ ) s , which has been widely used in different subjects, for instance, the thin obstacle problem, ecology, finance, and anomalous diffusion. For a comprehensive discussion of the properties and applications of the fractional Laplacian ( − Δ ) s for more details, the readers can refer to the guide [19]. Here, the potential functions A , V satisfy

1. A ∈ C 0 , μ ( R N , R N ) with μ ∈ ( 0 , 1 ] is a Z N -periodic vector potential. That is, for all x ∈ R N , it holds that A ( x + y ) = A ( x ) , ∀ y ∈ Z N ;

2. V ∈ C ( R N ) ∩ L ∞ ( R N ) , V ( x ) ⩽ V ¯ ≔ max x ∈ R N V ( x ) ∈ ( 0 , ∞ ) , and there exists a constant ζ 0 > 0 such that

   [ u ] A ε 2 + ∫ R N V ε ( x ) ∣ u ∣ 2 d x ⩾ ζ 0 ∫ R N [ V ¯ − V ε ( x ) ] ∣ u ∣ 2 d x for all u ∈ H ε s ,

   where A ε ( x ) ≔ A ( ε x ) and V ε ( x ) ≔ V ( ε x ) ;

3. V ( x ) = V P ( x ) − W ( x ) , where V P ∈ C ( R N , R ) is Z N -periodic, and W ∈ L N 2 s ( R N , R ) with W ( x ) > 0 , ∀ x ∈ R N ;

1. f ∈ C ( R N + 1 , R ) , and there exist C 0 > 0 and p 1 , p 2 with 2 N − α N < p 1 ⩽ p 2 < 2 s , α ∗ , such that for any t ∈ R

   ∣ f ( x , t ) ∣ ⩽ C 0 ∣ t ∣ p 1 − 2 2 + ∣ t ∣ p 2 − 2 2 uniformly in x ∈ R N ;

2. t ↦ f ( x , t ) is nondecreasing on R for every x ∈ R N ;

3. lim ∣ t ∣ → + ∞ F ( x , t ) ∣ t ∣ 2 s , α ∗ − 2 s N − 2 s 2 = + ∞ uniformly in x ∈ R N ,

   where F ( x , t ) = ∫ 0 t f ( x , s ) d s ;

4. f ( x , t ) is Z N -periodic with respect to x ∈ R N .

The following nonlinear Choquard equation

seems to arise from the work of Pekar [34] for the modeling of quantum polaron and was mentioned in [26] that Choquard used this equation to study a certain approximation to Hartree-Fock theory of one component plasma. When α = 1 , p = 2 , V ∈ L ∞ ( R 3 ) is Z 3 -periodic and 0 lies in the gap of the spectrum of − Δ + V ( x ) , Buffoni et al. [10] proved the existence of a nontrivial solution for (1.2) by using variational methods and Lyapunov-Schmidt reduction, and Ackermann [1] obtained the existence and multiplicity of solutions for (1.2) by applying an abstract critical point theorem. Wu et al. [39] considered the following Choquard equation with lower critical exponent:

where α ∈ ( 0 , N ) , N ⩾ 1 and V satisfies ( V 1 ) and ( V 2 ) . In their paper, the condition ( V 2 ) introduced in [18] is applied to obtain an equivalent norm of H 1 ( R N ) . The nonlinear perturbation f satisfies suitable growth including ( F 4 ) . The authors proved that there exists a ground state solution of above problem by applying variational methods and Lions’ concentration compactness principle. Bueno et al. [9] considered the following magnetic Choquard equation with upper critical exponent

where α ∈ ( 0 , N ) , 2 α ∗ = 2 N − α N − 2 , p ∈ 2 N − α N , 2 α ∗ , N ⩾ 3 , λ > 0 , A and V satisfy conditions ( A ) and ( V 2 ) , respectively. The authors obtained the existence of at least one ground state solution for the aforementioned problem, provided that p belongs to some intervals that depend on N and λ . Problems like or similar to (1.2) have been extensively studied in recent years by many authors, and the readers can refer to [6,11,12,14,20,21,28,25,30,31,33,35] and the references therein.

On the other hand, nonlinear Choquard equation of the type

has also received much attention recently. When s = 1 , α ∈ ( 0 , N ) , N ⩾ 3 , and G ( x , t ) = ∣ t ∣ p with p ∈ 2 N − α N , 2 α ∗ , Moroz and Van Schaftingen [32] testified that (1.3) has a family of solutions concentrating to the local minimum of V for small ε by a novel nonlocal penalization technique, provided that the external potential V ∈ C ( R N , [ 0 , ∞ ) ) satisfies some additional assumptions at infinity. When s = 1 , N = 3 , α ∈ ( 0 , 3 ) , G ( x , t ) = Q ( x ) H ( t ) and H has a critical growth, Alves et al. [3] obtained the existence and multiplicity of solutions for (1.3) and characterized the concentration behavior for small ε , provided that Q , V ∈ C ( R 3 , ( 0 , ∞ ) ) satisfy some additional assumptions at infinity. When s ∈ ( 0 , 1 ) , N > 2 s , α ∈ ( 0 , 2 s ) , V is a continuous potential function and satisfies the following local conditions, which was introduced in [17]:

1. inf x ∈ R 3 V ( x ) > 0 , and there is a bounded open domain Ω such that

   V 0 ≔ inf Ω V ( x ) < min ∂ Ω V ( x ) ,

Motivated by the works [9,39,40], in the present paper, we focus our attention on the existence of ground state solutions for problem (1.1). We emphasize that the lack of compactness due to the fact that (1.1) contains the upper critical exponent 2 N − α N − 2 s in the sense of Hardy-Littlewood-Sobolev inequality and the nonlocal nature of the fractional magnetic operator bring the main difficulties. To handle these difficulties, we first consider the problem (1.1) with periodic electric potential, that is, V = V P , and prove that the energy functional has a mountain pass geometry. Thus, we obtain a Cerami sequence at the mountain pass level (denoted by c ε ) by [22], and we can show the boundedness of this sequence by using the monotone condition ( F 2 ) . Next, we prove the existence of ground state solutions for problem (1.1) with periodic electric potential V P (see Theorem 3.5) by using a fractional version of the Lions’ concentration compactness principle ([36, Lemma 2.4]), the estimate of the level c ε (see Lemma 3.4) and the periodicity condition on A , V , and f . Finally, taking advantage of the estimate of c ε > d ε for all ε > 0 , where d ε is the mountain pass level of the energy functional associated with the original problem (1.1), we get the existence of ground state solutions for (1.1) by applying the concentration compactness argument once again. The main result can be stated as follows.

The paper is organized as follows. In Section 2, we introduce the functional setting and we recall some useful lemmas for the fractional magnetic Sobolev spaces. In Section 3, we give the existence result for (1.1).

2 Preliminaries

In this section, we are devoted to some notations and preliminary results. Let D s , 2 ( R N , R ) denote as the closure of C c ∞ ( R N , R ) with respect to

The fractional Sobolev space H s ( R N , R ) is defined as follows:

endowed with the norm ‖ u ‖ 2 ≔ [ u ] s 2 + ‖ u ‖ L 2 2 ; see [19] for more details.

The space D A s , 2 ( R N , C ) is defined as follows:

where

The fractional magnetic Sobolev space H ε s is defined as follows:

under the scalar product

with the associated norm ‖ u ‖ H ε s 2 = ⟨ u , u ⟩ H ε s , where Re ( w ) denotes the real part of w ∈ C and w ¯ denotes its complex conjugate.

Lemma 2.1 implies that

is an equivalent norm of H ε s with the scalar product

Referring to [8,16], we recall the following useful properties for the space H ε s .

We use S H , L to denote the best constant defined by

From Lemma 2.2, we know that

Lieb and Loss [27] introduced the following well-known Hardy-Littlewood-Sobolev inequality.

3 Existence of ground state solution

In this section, we study the existence of ground state solution for problem (1.1). Using the change of variable u ( x ) ↦ u ( ε x ) , we can see that problem (1.1) is equivalent to

Initially, we consider the following nonlocal problem with periodic potential associated with (3.1)

where V P , ε ( x ) ≔ V P ( ε x ) .

Combining ( V 1 ) with ( V 2 ) , by a similar discussion as in Lemma 2.1, we get that

is an equivalent norm of H ε s with the scalar product

The associated energy functional J P , ε : H ε s → R for problem (3.2) is defined as follows:

Under our assumptions and using Lemma 2.5, we get that J P , ε is well defined on H ε s and belongs to C 1 ( H ε s , R ) . Thus, u is a weak solution of problem (3.2) if and only if u is a critical point of the functional J P , ε . Next, we prove that J P , ε has the mountain pass geometry.

Let c ε denote the mountain pass level of J P , ε by

where Γ ε ≔ { γ ∈ C ( [ 0 , 1 ] , H ε s ) : γ ( 0 ) = 0 , J P , ε ( γ ( 1 ) ) < 0 } , and we have the following characterization of the mountain pass level c ε of J P , ε .