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Journal of Function Spaces
Volume 2018, Article ID 7935706, 5 pages
https://doi.org/10.1155/2018/7935706
Research Article

Non-Nehari Manifold Method for Fractional p-Laplacian Equation with a Sign-Changing Nonlinearity

1Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004, China
2College of Information Sciences and Technology, Hainan University, Haikou 570228, China
3School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, China

Correspondence should be addressed to Shengjun Li; moc.621@626iljhs

Received 15 April 2018; Accepted 10 June 2018; Published 18 July 2018

Academic Editor: Xinguang Zhang

Copyright © 2018 Huxiao Luo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider the following fractional p-Laplacian equation: , where , , , is the fractional -Laplacian, and and for a.e. . has the subcritical growth but higher than ; however, the nonlinearity may change sign. If is coercive, we investigate the existence of ground state solutions for p-Laplacian equation.

1. Introduction

Consider the following nonlinear Schrödinger equation with fractional -Laplacian:where , , , and is the fractional -Laplacian. , , and satisfy the following assumptions:(), there exists a constant such that where meas denotes the Lebesgue measure in ;() for a.e. ;() is measurable, continuous in for a.e. and there are and such that where ;() as uniformly in ;() uniformly in as , where ;() is nondecreasing on .

When , (1) arises in the study of the nonlinear Fractional Schrödinger equation Problems with this type have occurred in many different fields, such as continuum mechanics, phase transition phenomena, population dynamics, and game theory, as they are the typical outcome of stochastically stabilization of Lévy processes; see [14].

When and , (4) reduces to be the nonlinear Schrödinger equation Using the Nehari-type monotonicity condition, Szulkin and Weth [5] obtained the existence of ground state solutions for (5). But in this paper, the Nehari manifold is usually not smooth and the Nehari-type monotonicity condition for the nonlinearity is not satisfied; then the Nehari manifold method is invalid. In this paper, we are aimed to obtain ground state solutions for (1) by the so-called non-Nehari manifold method, which is established by Tang [6, 7]. Unlike the Nehari manifold method, the main idea of our approach lies on finding a minimizing sequence for the energy functional outside the Nehari manifold by using the diagonal method.

Now, we are ready to state the main result of this paper.

Theorem 1. Suppose that and hold. Then (1) has a nontrivial ground state solution.

2. Preliminaries

In the paper, we will denote by the infinitesimal as . For the sake of simplicity, the norm of the space will be denoted by , and integrals over the whole will be written .

We define the Gagliardo seminorm by where is a measurable function. Then fractional Sobolev space is given by endowed with the norm For the basic properties of fractional Sobolev spaces, we refer the interested reader to [8]. By condition , we define the fractional Sobolev space with potential by endowed with the norm The energy functional defined by Under our hypotheses, is well defined on . It is well known that , and its derivative is given by for . It is standard to verify that the weak solutions of (1) correspond to the critical points of . Now, we review the main embedding result for the space .

Lemma 2 ([9, Lemma 1]). Under assumption , the embedding is compact for any .

In the following lemma, we will show that has Mountain Pass geometric structure.

Lemma 3. Suppose that , and hold.
(i) There is such that .
(ii) For any , there exists such that .

Proof. (i) By and , we haveBy for and (13), we have By the arbitrariness of and , we get the conclusion.
(ii) Fix ; by , we have as . Thus, there exists such that .

Now, we define the Nehari manifold by It is easy to prove that is not empty. And we have the following lemma.

Lemma 4. Suppose that and hold. Let and ; then .

Proof. By , we have ThenLet ; then . By a simple calculation, we have and for all . Thus,Let , it follows from (18) and (19) that

Lemma 5. Suppose that and hold; let ; then there exist satisfying

Proof. By (i) of Lemma 3, there exist and such that Choose such thatSince for large , Mountain Pass Lemma implies that there exists satisfyingwhere . By Lemma 4, we have Hence, by (23) and (24), we have Now, we can choose a sequence such that Let . Then, going if necessary to a subsequence, we have

3. Proof of Theorem 1

Proof of Theorem 1. In view of Lemma 5, we find a Cerami sequence satisfying (21). By (18), we haveCombining (21) and (28), for big enough, we have It follows that is bounded. Passing to a subsequence, we have in . By Lemma 2, we have in for . Then, by (13) and the Hölder inequality, we haveandIt follows from (30), (31) and Simon inequality () thatOn the other hand, by and , we haveCombining (32) and (33), we have in . Then, by , we have . By (28), Lemma 5, and Fatou’s lemma, we have This shows that and so .

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

Acknowledgments

This work is supported by the Hainan Natural Science Foundation (Grant nos. 118MS002 and 117005), National Natural Science Foundation of China (Grant nos. 11461016 and 11571370), China Postdoctoral Science Foundation Funded Project (Grant no. 2017M612577), and Young Foundation of Hainan University (Grant no. hdkyxj201718).

References

  1. D. Applebaum, “Lévy processes—from probability to finance and quantum groups,” Notices of the American Mathematical Society, vol. 51, no. 11, pp. 1336–1347, 2004. View at Google Scholar · View at MathSciNet
  2. N. Laskin, “Fractional quantum mechanics and Lévy path integrals,” Physics Letters A, vol. 268, no. 4–6, pp. 298–305, 2000. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  3. R. Metzler and J. Klafter, “The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics,” Journal of Physics A: Mathematical and General, vol. 37, no. 31, pp. R161–R208, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  4. B. Bieganowski, “Solutions of the fractional Schrödinger equation with a sign-changing nonlinearity,” Journal of Mathematical Analysis and Applications, vol. 450, no. 1, pp. 461–479, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  5. A. Szulkin and T. Weth, “Ground state solutions for some indefinite variational problems,” Journal of Functional Analysis, vol. 257, no. 12, pp. 3802–3822, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. X. H. Tang, “Non-Nehari manifold method for superlinear Schrödinger equation,” Taiwanese Journal of Mathematics, vol. 18, no. 6, pp. 1957–1979, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  7. X. H. Tang, “Non-Nehari manifold method for asymptotically periodic Schrödinger equations,” Science China Mathematics, vol. 58, no. 4, pp. 715–728, 2015. View at Publisher · View at Google Scholar
  8. E. Di Nezza, G. Palatucci, and E. Valdinoci, “Hitchhiker's guide to the fractional Sobolev spaces,” Bulletin des Sciences Mathématiques, vol. 136, no. 5, pp. 521–573, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  9. V. Ambrosio, “Multiple solutions for a fractional p-Laplacian equation with sign-changing potential,” Electronic Journal of Differential Equations, vol. 2016, article 151, 12 pages, 2016. View at Google Scholar