Small perturbations of critical nonlocal equations with variable exponents

1 Introduction

In this article, we deal with the existence of solution to the following critical nonlocal equations with variable exponents:

where Ω ⊂ R N is a bounded domain with Lipschitz boundary, N ≥ 2 , p ∈ C ( Ω × Ω ) is symmetric, almost everywhere, p ( x , y ) = p ( y , x ) for all ( x , y ) ∈ R N × R N , q ∈ C ( Ω ) satisfies

for all x ∈ R N and λ is a real parameter. The functional f ( x , t ) ∈ C ( Ω × R ) satisfies the following growth assumptions:

1. lim t → 0 f ( x , t ) ∣ t ∣ p ( x ) − 2 t = 0 uniformly in x ;

2. lim t → ∞ f ( x , t ) ∣ t ∣ p − − 2 t = + ∞ uniformly in x .

Our interest in Problem (1.1) is mainly based on theoretical and practical reasons. On the one hand, many scholars pay attention to the study of differential equations and variational issues involving p ( x ) -growth conditions in recent years. The development of numerous significant models in electrorheological and thermorheological fluids, image processing, and other fields inspired a systematic study of partial differential equations with variable exponents (see [1–4]). As we all know, Lebesgue spaces with variable exponents were initially studied in this article [5] by Orlicz. Then, we can obtain more details on Lebesgue Sobolev spaces related to the p ( ⋅ ) -Laplacian with variable exponents in [6,7]. The Lebesgue-Sobolev spaces related to the p ( ⋅ ) -Laplacian are called variable-exponent Lebesgue-Sobolev spaces and were studied in [8,9]. We also refer the reader to the book [10] for basic knowledge in this area. The literature on the study of such operators is large but we only list a few of them and the recently published articles for interested readers (see e.g., [11–18]). In these articles, the authors used different methods to establish the existence of solutions and some other properties under the subcritical growth condition. Very recently, Cao et al. in [19] established the existence of nontrivial solutions for p ( x ) -Laplacian equations

where Ω ⊂ R N is a bounded domain with Lipschitz boundary, p ∈ C ( Ω ¯ ) with 1 < p ( x ) < N , q ∈ C ( Ω ¯ ) with p + ≔ max x ∈ Ω ¯ p ( x ) < q − = min x ∈ Ω ¯ q ( x ) ≤ q + ≔ max x ∈ Ω ¯ q ( x ) < p ∗ ( x ) , p ∗ ( x ) = N p ( x ) N − p ( x ) is the critical exponent and f : Ω × R → R is a continuous function without any growth and Ambrosetti-Rabinowitz conditions. However, the study of critical problem is a very meaningful subject, and the critical problem was initially studied in the seminal paper of Brézis and Nirenberg in [20], which treated for Laplace equations. Since then, there have been extensions of [20] in many directions. Elliptic equations involving critical growth are delicate due to the lack of compactness arising in connection with the variational approach. For such problems, the concentration-compactness principles introduced by Lions in [21,22] and its variants at infinity in [23,24] have played a decisive role in showing that a minimizing sequence or a Palais-Smale sequence is precompact. By using these concentration-compactness principles or extending them to the Sobolev spaces with variable exponents, many authors have been successful to deal with critical problems involving p ( ⋅ ) -Laplacian (see e.g., [24–29] and references therein).

On the other hand, nonlocal operators can be seen as the infinitesimal generators of Lévy stable diffusion processes in [30]. Moreover, they allow us to develop a generalization of quantum mechanics and also to describe the motion of a chain or an array of particles that are connected by elastic springs as well as unusual diffusion processes in turbulent fluid motions and material transports in fractured media (for more details, see, for example, [31,32] and references therein). The study of elliptic equations with nonlocal operators is one of the most fascinating areas of nonlinear analysis. These issues have received a lot of attention in both pure mathematics study and practical applications (see, for example, the recent monograph [33]). The study of fractional Sobolev space with variable exponents is natural extension of variable-exponent Lebesgue Sobolev spaces. As far as we know, the fractional Sobolev spaces with variable exponent and the fractional p ( x ) -Laplacian were introduced first by Kaufmann et al. in [34]. Here, the authors obtained the embedding result of these spaces to variable-exponent Lebesgue spaces. In addition, they also discussed the existence result of a fractional p ( x ) -Laplacian problem. Very recently, Ho and Kim obtained fundamental embedding results for the new fractional Sobolev spaces with variable exponents that are a generalization of the well-known fractional Sobolev spaces in [35]. Using this, they demonstrated a priori bounds and multiplicity of solutions of some nonlinear elliptic problems involving the fractional p ( ⋅ ) -Laplacian. We refer to [36–38] for fractional Sobolev spaces with variable exponents and the corresponding nonlocal equations with variable exponents.

Inspired by the works in the aforementioned references, our main purpose in this article is to study the existence of solution for Problem (1.1). Moreover, we do not assume that f satisfy the well-known Ambrosetti-Rabinowitz-type condition, and q ( x ) can reach the critical exponent p s ∗ ( x ) . Therefore, it can be viewed as a partial extension of the results of Cao et al. in [19] concerning the existence of solutions to equation (1.2) in the case of s = 1 and subcritical case.

We are ready to state the main result of this article.

The main difficulty in treating Problem (1.1) is the possible lack of compactness due to the presence of critical term and the growth condition of function f . To overcome the difficulties that arise from these features, we use the cutoff function approach and the concentration-compactness principles for fractional Sobolev spaces with variable exponents to prove that auxiliary problem has at least one nontrivial solution. Finally, we obtain nontrivial solutions for original problems with the aid of the Moser iteration method.

The rest of this article is organized as follows. In Section 2, we present some necessary preliminary knowledge on variable-exponent Lebesgue space and fractional Sobolev space with variable-exponent space. In Section 3, we prove the Palais-Smale condition at some special energy levels by using the concentration-compactness principles for fractional Sobolev spaces with variable exponents. In Section 4, we prove the uniform bound of weak solution to auxiliary problem. Section 5 is devoted to proving the existence of nontrivial solutions of Problem (1.1) by the Moser iteration method.

2 Preliminaries

In this section, we briefly review the definitions and list some basic properties of the Lebesgue spaces. Furthermore, we recall and establish some qualitative properties of the new fractional Sobolev spaces with variable exponents.

3 An auxiliary problem

In this section, we mainly prove that the energy functional associated with Problem (1.1) satisfies the ( P S ) c at some special energy levels. Due to assumption ( f 2 ) , we need to truncate the function f ; thus, there is a constant Q > 0 that is large sufficiently so that f ( x , Q ) > 0 . For all x ∈ Ω , let

where p + < q − ≤ q ( x ) ≤ q + . Then, because of the continuity of f , we claim that the cutoff function f Q : Ω × R → R is continuous. Moreover, by assumption ( f 1 ) and (3.1), we know that

1. lim t → 0 f Q ( x , t ) t p ( x ) − 1 = 0 uniformly in x ;

2. lim t → ∞ F Q ( x , t ) t p + = + ∞ uniformly in x , where F Q ( x , t ) = ∫ 0 t f Q ( x , s ) d s ;

3. ∣ f Q ( x , t ) ∣ ≤ C Q 1 ∣ t ∣ p ( x ) − 1 + C Q 2 ∣ t ∣ q ( x ) − 1 for all ( x , t ) ∈ Ω × R , where

   C Q 1 = max x ∈ Ω ¯ , t ∈ [ 0 , Q ] f ( x , t ) t p ( x ) − 1 and C Q 2 = 1 Q q − − 1 max x ∈ Ω ¯ f ( x , Q ) ;

4. for any θ ∈ ( p + , q − ) , there exists τ = τ ( Q ) > 0 such that

   1 θ t f Q ( x , t ) − F Q ( x , t ) ≥ − τ t p ( x ) for all ( x , t ) ∈ Ω × [ 0 , + ∞ ) .

The aim of this section is to prove the existence of nontrivial weak solution for Problem (3.2), that is, u ∈ W s , p ( ⋅ , ⋅ ) ( Ω ) is a solution of Problem (3.2) if

where

for all v ∈ W s , p ( ⋅ , ⋅ ) ( Ω ) . Problem (3.2) has a variational nature, and the energy functional J λ Q : W s , p ( ⋅ , ⋅ ) ( Ω ) → R associated with it is defined as follows:

where

and F Q ( x , t ) = ∫ 0 t f Q ( x , s ) d s . It is easy to check that J λ Q ∈ C 1 ( W s , p ( ⋅ , ⋅ ) ( Ω ) , R ) . Therefore, all solutions of Problem (3.2) correspond to critical points of the functional J λ Q in the weak sense.

Now, we prove that functional J λ Q has the mountain pass geometry.

It follows from Lemma 3.1 that there exists a ( P S ) c λ Q sequence { u n } ⊂ W s , p ( ⋅ , ⋅ ) ( Ω ) such that

at the minimax level

where Γ = { γ ∈ C 1 ( [ 0 , 1 ] , W s , p ( ⋅ , ⋅ ) ( Ω ) ) : γ ( 0 ) = 0 , J λ Q ( γ ( 1 ) ) < 0 } .

Furthermore, we can obtain the following compactness result.