The Nehari Manifold for a Fractional p-Kirchhoff System Involving Sign-Changing Weight Function and Concave-Convex Nonlinearities

Abstract

In this paper, we are concerned with the following fractional p-Kirchhoff system with sign-changing nonlinearities: , , and , where is a smooth bounded domain in , , , , are two real parameters, , is a continuous function, given by , , , are sign changing and either or , with , and . Using Nehari manifold method, we prove that the system has at least two solutions with respect to the pair of parameters .

1. Introduction

In this article, we study the following fractional p-Kirchhoff problem involving concave-convex nonlinearities and sign-changing weight functions: where is a smooth bounded domain in , , , , are two real parameters, , is a continuous function, given by , , , , and is the fractional p-Laplacian operator, which is defined as follows:where , , , satisfy the following assumptions: set ,

  , and either or ;

   with and .

In recent years, the Kirchhoff equations have received extensive attention from many scholars because of its wide application in many fields such as mathematical finance, continuum mechanics, etc. (see [1, 2]). There are many excellent and interesting results about the existence and multiplicity of solutions for nonlocal fractional problems. We can look up the literature [3, 4] for Laplace operator and [5–8] for the p-Laplacian case.

In addition, for a single equation with sign-changing weights functions, in [9, 10], the authors studied the existence and multiplicity of nonnegative solutions in subcritical and critical cases respectively. In the special case of , , and , Tsung-Fang Wu [11] proved that system has least two nontrivial nonnegative solutions by using the Nehari manifold. Moreover, when is not a constants, the authors [12] investigated the fractional p-Kirchhoff system with sign-changing nonlinearities, which is given by . For a more general case , Yang and An [13] show the system has at least two solutions with the help of Nehari manifold, but without considering sign-changing weights functions. Hence, inspired by above works, combining [12, 13], in this paper we will consider the new multiplicity result of the problem. Our conclusions can be seen as an extension of [12, 13].

To illustrate our result, we need to introduce some notations. Set be an open set in , , and . Define the usual fractional Soblev Space and its normLet and . Define the space as Then the space of norm is defined by

Set Banach space to be the completion of the space in , which is can be defined as the norm

Clearly, (6) is equivalent to the (3), as a.e. in , we obtain that the integral in (3), (5), and (6) can be extended to the full space . According to the literature [14, 15], we know that is a continuous embedding for any , and compact whenever . When , then, for any , we get that

More about the properties of and , please consult [16] and the references therein. The reflexive Banach space is the Cartesian product of two spaces, which is endowed with the norm

Definition 1. A pair of functions is called weak solution of problem if for all one has

We introduce the setThen we give our result as follows.

Theorem 2. Assume that the conditions and hold. If , then there is an explicit constant such that the problem has at least two nonnegative solutions for .

The rest of the paper is organized as follows. In Section 2, we give some notations and preliminaries about Nehari manifold and fibering maps. In Section 3, we prove Theorem 2.

2. The Variational Setting and Preliminaries

Define energy functional associated with problem (P) as follows:where and and

By a direct calculation we obtain that , and for all , we haveThen, the weak solution is equivalent to being a critical point of . Since is not bounded below on , therefore, we consider the Nehari manifoldBy (13), we getHence, if and only ifMoreover, for any , the following equality holds:Obviously, the solution of the problem depends on . is a set which we find that is smaller than , so it is easier to study on . Therefore, define our familiar fiber maps: as follows:It follows from (19) that if and only if . So it is natural that we divide into three parts: local minima, local maxima, and points of inflection. For this, we let

DefineMoreover, for every , from (16), we also haveNow, we give some preliminaries about main result.

Lemma 3. If is a minimizer of on and . For every , then we have in .

Proof. For the detailed process of certification we can refer to the literature ([17], Theorem 2.3). For the convenience of the reader we give its completeness. If is a local minimizer on to , by the theory of Lagrange multipliers, there is a constant such thatSoBut , because of . Therefore . This completes the proof.

Lemma 4. is coercive and bounded below on .

Proof. By the Sobolev inequalities and Hölder inequalities, we getAlso, according to (17) and (26), we have As , from above inequality, we can conclude is coercive and bounded below on . We complete this proof.

Lemma 5. Under condition , there exists , given bysuch that, for any , we have .

Proof. We argue by contradiction, moreover dividing the following two cases: assume that there exist such that . Then for , we have

Case 1. . From (29), (19), and (20), we havewhich is a contradiction.

Case 2. , then it follows from (29), (23), and (26) thatIn addition, by condition and Young’s inequality, we getBy (29), (32), and (23), we haveand hence, we obtainFrom (31) and (34), we getwhich contradicts . We have completed the proof of this lemma.

According to Lemma 4 and Lemma 5, we know for , and we set We introduce the following lemma.

Lemma 6. Assume that . Then, we have
(i) ,
(ii) for some .

Proof. (i) Set , we know , and from (23), we havePut (37) into (17),which implies .
(ii) It follows from (17) and (23) that Since , then . Hence, according to (17), we obtain (34), combining above inequality and (34), and we get Obviously, if , then there is a constant such that .

Under condition , we study the behavior of the fibering map with respect to the sign of . We divide into two cases.

Case 1 (). Fix . LetClearly, if and only if .It is obvious that as . By (42), we know thatThus, there is a unique , such that is increasing on , decreasing on . Moreover, , in addition,where is the root ofFrom above the equality, we obtainSince , according to (26) and (46), we get Hence, there are unique and such that . It means that and . Moreover it is easy to see that , also and imply and . Since , then for any and for any . Thus, Furthermore, for any , , and for any imply that

Case 2 (). As we know that as , therefore for any , there is a unique such that ; moreover for any and for any , which implies that and .

Thus according to the above discussion we obtain that the following lemma.

Lemma 7. Under condition , for any , we have the following:
(i) If , then there is a unique such that , and (ii) If , then there exist a unique and unique such that , , and

3. Proof of the Main Result

In this section, we establish the existence of minimizers in and

Proposition 8. Under condition , if , then the functional has a minimizer in and fulfills the following:
(i) ,
(ii) is a solution of problem .

Proof. Since is bounded from below on , there is a minimizing sequence such thatHence, by Lemma 4, then is bounded on . So there exists , up to a subsequence, such thatMoreover, according to ([3], lemma 8),For each , by ([18], Theorem IV-9), there exists such thatHence, by the dominated convergence theorem, we getandas . Now, on , from (17), we haveLetting , since , from Lemma 6, (50), and (54), we getFrom Lemma 7, there exists such that and Next we show that strongly in . If this is not true, thenSince andwe obtainIt means that for large enough. As , it is obvious to know that , and for . Thus we have . On the other hand, is decreasing on , sowhich is a contradiction. Thus strongly in . This meansThat is, is a minimizer of on ; using Lemma 3, is a solution of problem .

Proposition 9. Under condition , if , then the functional has a minimizer in and fulfills the following:
(i) ,
(ii) is a solution of problem .

Proof. is bounded from below such thatIt is similar to the proof of the Proposition 8, so there exists , up to a subsequence, such thatMoreover,For each , by the dominated convergence theorem, we also get andand similarly, by Lemma 7, there is a such that . Next we show that strongly in . Suppose that this is not true, thenSince and for any , we also havewhich is a contradiction. Hence strongly in . This impliesNamely, is a minimizer of on ; by Lemma 3, is a solution of problem .

Proof of Theorem 2. According to Propositions 8 and 9, we obtain that, for , problem has two solutions and in . Sincemoreover , so we get that are distinct nonnegative solutions.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The work is supported by the Fundamental Research Funds for Central Universities (2017B19714 and 2017B07414), Natural Science Foundation of Jiangsu Province (BK20180500), and Natural Science Foundation of Jilin Engineering Normal University (XYB201814). The work is also supported by Program for Innovative Research Team of Jilin Engineering Normal University.