Ground State Solutions to a Critical Nonlocal Integrodifferential System

Abstract

Consider the following nonlocal integrodifferential system: , where is a general nonlocal integrodifferential operator, , is a fractional Sobolev critical exponent, , , is a lower order perturbation of the critical coupling term, and is an open bounded domain in with Lipschitz boundary. Under proper conditions, we establish an existence result of the ground state solution to the nonlocal integrodifferential system.

1. Introduction

Recently, fractional Sobolev spaces and the corresponding nonlocal equations are applied to many subjects, such as, anomalous diffusion, elliptic problems with measure data, gradient potential theory, minimal surfaces, nonuniformly elliptic problems, optimization, phase transitions, quasigeostrophic flows, singular set of minima of variational functionals, and water waves (see [1] and the references therein). For more details, please refer the book [2]. In [3], the authors considered the following equation:where is a general nonlocal integrodifferential operator of order which will be defined later and is a lower order perturbation of the critical power , and, under proper conditions, an existence result of its solutions was obtained.

In present paper, we use some ideas and techniques in [3–5] to extend the results in [3] to the system case.

Consider the following nonlocal integrodifferential system:where , is a fractional Sobolev critical exponent, , , is a lower order perturbation of the critical coupling term, is an open bounded domain in with Lipschitz boundary, and is a general nonlocal integrodifferential operator defined as Here, is a function satisfying

Define the Hilbert space as the completion of with respect to the norm induced by the scalar product given by If is an open bounded Lipschitz domain, then coincides with the Sobolev spacewhere is a linear space of Lebesgue measurable functions from to such that the restriction to of any function in belongs to and the map is in , and is the complement of in .

The solutions of (2) coincide with the critical points of the following energy functional :where , , , endowed with norm , and is a nonnegative Carathéodory function from to ; namely,Consider the following Nehari manifold: define the ground state energy of (2) byand call a solution by a ground state solution if . DefineThen , where

Theorem 1. Assume that , where is the first eigenvalue of the nonlocal operator with homogeneous Dirichlet boundary conditions. If there exists with a.e. in , such thatthen (2) has a nontrivial ground state solution.

2. Preliminaries

We need the following results, which have been proved in [6].

Lemma 2. There exists a positive constant such that, for any , where is only depending on and .

3. Proof of Theorem 1

Lemma 3. Assume that is a Carathéodory function; that is, (10)–(12) hold. Then, for any , there exist and such that for a.e. and, for any ,

Proof. For any , by (12), there exists such that, for a.e. and for any with , It follows from (10) that there exists satisfying, for a.e. and for any with ,Combining (21) and (22), we see that (19) holds.
By (11), for any , there exists such that, for a.e. and for any with , It is easy to see from (19) that Therefore, there exists satisfying, for a.e. and for any with ,Then, (20) follows from (23) and (25).

Lemma 4. Assume that . Then, there exist and such that (i) for any with ;(ii) a.e. in , , and . In particular, if (17) holds, then we can construct by , where is given by (17) and is large enough.

Proof. For and any , by (20), Lemma 2, , and (5), we deduce that For , we have where and are suitable positive constants. Let satisfy , where is small enough such that . Therefore, which proves (i).
Fix with and a.e. in . Choosing in (19), we get thatFor , by (29), we have which implies that as . Then, (ii) follows by taking with sufficiently large.
In particular, if (17) holds, then we may fix by . Thus, we can construct by , where is large enough.

Definewhere with given in Lemma 4. It is standard to see thatwhere and is given in (14).

Lemma 5. , where is given in Lemma 4 and is defined in (15).

Proof. Noting that, for any , the function is continuous in ; we get that and , where is given by Lemma 4. Thus, we may find such that . Therefore, which implies that . Recalling that and the special pass belongs to , by (17), we deduce that

Lemma 6. Every sequence with and in as ( sequence of ) is bounded in .

Proof. From and it can be seen thatfor some suitable positive constant . For , it follows from Lemma 2 and (5) thatfor some positive constant . By (19) and (39), we havewhere is a positive constant relevant to and . Choosing small enough such that , then (38) and (40) yield that Letting in (40) and noting , by (41), we infer that for suitable constants and , which implies that is bounded in .

Proof of Theorem 1. It follows from that , where is given in Lemma 4. By Lemma 4, we see that has a Mountain Pass Structure (e.g., [7]). By (33) as well as Lemma 5 and Theorem 2.2 in [8], there is sequence for . By Lemma 6, we get that is bounded in , and then, going if necessary to a subsequence, still denoted by , there exists such that weakly in ; i.e., for any . It follows from (41) and the boundedness of in thatThen, it is standard to see thatup to a subsequence, where . By (19), (44), and the boundedness of , we see that Since is superlinear with respect to and , we get that the mappings and are continuous in . Therefore, Then, by (46) and (47), we have Noting that is the dual space of , we deduce that and then, in particular, For any , we have Then, it follows from (43), (45), and (50) that which means that is solution of (2). It remains to show that is nontrivial. Suppose by contradiction that in . Then, by (19) and boundedness of and in , we have and similarly Noting that in for any and letting , we get that Taking , we have Therefore, Assume, up to a subsequence, that . Then, , and which means that . By Lemma 5, we see that . Noting (15), we have which implies that . Therefore, , contradicting with Lemma 5. Hence, can not be . Besides (45), up to a subsequence, we may assume that where is the power sum of with respect to and . From we get that where ; i.e., in . Setting and , then weakly in . Thus, strongly in , a.e. in , , , in , up to a subsequence. Brezis-Lieb Lemma guarantees that Since and , by (63), we deduce thatandwhere Assume that up to a subsequence. By (64), we see that Letting , we haveSuppose by contradiction that , where . Then, is a nontrivial solution of . Hence, by (69), we deduce that , contradiction with Lemma 5. Therefore, can not be , where . Similarly, can not be , where . Thus, (2) has a nontrivial ground state solution, which completes the proof.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is supported by NSFC (11701248).