Multiplicity of Weak Positive Solutions for Fractional & Laplacian Problem with Singular Nonlinearity

Abstract

In this paper, we prove the existence and multiplicity of positive solutions for a class of fractional & Laplacian problem with singular nonlinearity. Our approach relies on the variational method, some analysis techniques, and the method of Nehari manifold.

1. Introduction

In this paper, we consider the following fractional & Laplacian problem

where is an open bounded domain with smooth boundary , , , , is a positive parameter. The weight functions is in with for almost every , is bounded with for almost every , and denotes the critical Sobolev exponent. , with , is the fractional -Laplacian operator defined for any by

Recently, Wang and Zhang [1] investigated the following singular elliptic boundary value problem involving the fractional Laplacian

the existence and multiplicity of weak positive solutions of (3) have been obtained in [1] by using the variational method, Nehari Maniford method, and Fibering Map analysis.

In [2], Crandall et al. firstly studied the semilinear problem with singular nonlinearity, since then, the local setting () for (3) and some other versions of the problem have been extensively studied during the past decades, see for example [3–10] and the references therein.

In recent years, the fractional Laplacian problems have been extensively investigated. For more details, we cite the reader to [11–15]. There are many different definitions of weak solutions for the fractional Laplacian equation (3). In [16], Fang say that is a weak solution of (3) with and if the identity

holds. In virtue of the method of sub-supersolution, the author gives the sufficient conditions for the existence and uniqueness of positive solution.

Very recently, great attention has been devoted to the study of fractional -Laplacian problems, see for instance [17–20]. However, in literature, there are only a few papers [21–23] dealing with fractional & problems. Motivated by the works [1, 23, 24], in this paper, we investigate the existence and multiplicity of solutions for the fractional & Laplacian problem (1) and extend the main results of Wang and Zhang [1].

This article is organized as follows. In Section 2, we give some notations and preliminaries. Section 3 is devoted to proving that problem (1) has at least two positive solutions for sufficiently small.

2. Preliminaries

For any , , we define

where with .

The space endowed with the norm

From [23], we know that , which allows us to study (1) in .

Throughout this section, we denote the best Sobolev constant for the embedding of into , defined as

By a weak solution of (1), we mean satisfies (1) weakly, that is, we are looking for a function , a.e. in such that

To study weak solutions to (1), we consider the following energy functional

The following Lemmas will be useful in the study of our problem (1).

Lemma 1 [23]. If , then with continuous embedding.

Lemma 2 [23]. If is a sequence weakly convergent to some in , then

The Nehari manifold is closed linked to the behavior of functions of the form for that named fibering maps [25]. If , we have

and

Obviously,

which implies that for and , if and only if , i.e., positive critical points of correspond to points on the Nehari manifold. In particular, if and only if . Hence, we define

Throughout this paper, we make the following assumptions: (H1) with for all almost every . (H2) is bounded for all almost every .

By (H2), we know that there exists such that for all almost every .

For , one has that . Thus, there exists a constant such that

Lemma 3. Assume that condition (H1) holds with . Then there exists such that , for each .

Proof. If , by the definition of and , we get thatBy (H1), (16), and the Hlder inequality and fractional Sobolev inequalities (7), we obtainIt follows from (17) and (18) thatThus, one hasThe proof is complete.

Lemma 4. If (H1) holds with , then the functional is coercive and bounded below on

Proof. By using the definition of and (16), we haveThis implies that is coercive and bounded from below on . The proof is complete.

Lemma 5. Assume that conditions (H1) and (H2) hold with . Then the minimal value .

Proof. For , applying (H2), the Hlder inequality and fractional Sobolev inequalities (7), we haveUsing inequality (18) and (22), we obtainIt follows from that there exist , , and small enough such that for any fixed where . On the other hand, for any fixed , since , we can derive thatwhere . Thus, for all , provided is sufficiently small. Hence, . This completes the proof of Lemma 5.

Lemma 6. For each , there exists such that .

Proof. Let be a minimizing sequence such that as . From Lemma 3 we know that the sequence is bounded in . Hence, we can obtain that there exists a sub-sequence of (still denoted by ) such that weakly in , strongly in () and pointwise a.e. in . By using Hlder inequality, we derive that as ,andThus, we obtain thatSince is bounded in , we have that is also bounded in . If and is a bounded sequence such that a.e. in , then we have from the Brezis–Lieb Lemma thatTakingin (29), we getSincewe can deduce that as By Lemmas 1 and 2, we haveandFrom , we have for some positive constant independent of . Thus, by (35) and we obtain if is sufficiently large. Combining above arguments with (28), (33), (34), and (35), we can obtainwhich yields thatPassing to the limit as , we get .
In the following, we will show that . It suffices to prove that strongly in . By andwe havewhich implies that as , i.e., strongly in . Thus, is a minimizer of in . The proof is complete.

3. Main Result

Similar to the proof of Lemma 3.7 in [1], we can easily obtain the following Lemma.

Lemma 7. For any , then there exists and a continuous function , with satisfying that , .

Lemma 8. For each given , , there exists such that for all and .

Proof. LetBy using the continuity of , and , we can infer that there exists , such that for all . On the other hand, applying Lemma 7 we get that for any there exists such that . Hence, as and for each we obtainThe proof of this Lemma is completed.

Lemma 9. The minimizer is a weak solution of problem (1). Moreover, for each .

Proof. Firstly, we prove that is a weak solution of (1). From Lemma 8, we derive that for any , and .Letting , we infer thatFrom Fatou’s Lemma, we haveTogether with (42) with (43), it yieldsFor any given , takinginto (44), we obtainwhere , and , . Since the measure of the set tends to 0 as , one hasHence, dividing by we infer thatSince is arbitrary, replacing by in above inequality, one obtainsThusHence, is a weak solution of (1).
Secondly, we prove that for almost each . Since , we get and . Let be the first eigenfunction of the operator with and . Taking in (50), one gets which implies that for almost every . This completes the proof of Lemma 9.

Lemma 10. Assume that (H1) and (H2) hold with . Then there exists such that for any , one has . Moreover, is closed in for all .

Proof. If not, there is a with . Then by the definition of , we getwhich implies thatand soOn the other hand, from the definition of , we can obtainBy using the Hölder inequalities and (7), we have by (57) thatwhich implies thatIf is sufficiently small, however, (55) contradicts to (58). Hence, we conclude that there exists such that for .
Let be a sequence satisfying in the . By using the Sobolev inequalities and continuous compact embedding, we have in and . From the definition of , we obtain thatthat is,which yields i.e., . The proof of this Lemma is completed.

Lemma 11. Assume that (H1) and (H2) hold with . Then there exists , such that , for each while .

Proof. If not, there is a such that . ThenBy (62) and the definition of , it follows thatCombining with (18) and (22), we getwhich implies thatWith the help of inequalities (61) and (65), for any , we get thatDirect calculations show thatwhich contradicts the fact as . We complete the proof of Lemma 11.

Lemma 12. Assume that (H1) and (H2) hold with . Then there exists small enough such that for each , there exists satisfying . Furthermore, is a weak positive solution of problem (1).

Proof. From Lemma 4, is coercive on , and it is also true for . If sequence satisfying as . By using the coercive of , we can get is bounded in . Hence, we may assume that weakly as in . Since is completed in (Lemma 10), it follows from the same arguments as in proving the existence of minimizer (Lemma 6) and the compactness of the embedding () we get is the minimizer of . Furtherover, similar to the proof of weak positive solution as in Lemma 9, one can prove that is also a weak positive solution for problem (1.1). The proof is complete.

Theorem 13. Let . Assume that (H1) and (H2) hold. Then there exists a positive number such that for each problem (1.1) possesses at least two weak positive solution ,.

Proof. Let . Obviously, Lemma 3 and Lemma 12 are true for all . Thus, it follows from Lemma 9 and Lemma 12 that and are the weak positive solutions of problem (1.1). The proof is complete.

4. Conclusions

In this paper, the existence and multiplicity of positive solutions for a class of fractional & Laplacian problem with singular nonlinearity have been investigated. It is worthy to point out that few studies have been done on this issue. By means of the variational method, Nehari manifold method and some analysis techniques, the sufficient conditions of existence and multiplicity of positive solutions to this problem have been presented in Theorem 13. Our results generalize the main conclusions of Wang and Zhang in [1].

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest.

Authors’ Contributions

The authors contributed equally to this paper. All authors read and approved the final manuscript.

Acknowledgments

The authors are thankful to the editor and anonymous referees for their helpful comments and suggestion. This work is supported by Natural Science Foundation of China (11571136).