About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2014 (2014), Article ID 809769, 7 pages
http://dx.doi.org/10.1155/2014/809769
Research Article

Three Weak Solutions for Nonlocal Fractional Laplacian Equations

Department of Mathematics, Huaiyin Normal University, Huaian, Jiangsu 223300, China

Received 16 September 2013; Revised 19 November 2013; Accepted 27 November 2013; Published 16 January 2014

Academic Editor: Salvatore A. Marano

Copyright © 2014 Dandan Yang and Chuanzhi Bai. 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

The existence of three weak solutions for the following nonlocal fractional equation in in is investigated, where is fixed, is the fractional Laplace operator, and are real parameters, is an open bounded subset of , , and the function satisfies some regularity and natural growth conditions. The approach is based on a three-critical-point theorem for differential functionals.

1. Introduction

In this work we investigate the existence of three weak solutions to the nonlocal counterpart of perturbed semilinear elliptic partial differential equations of the type namely, where is fixed, is a nonempty bounded open subset of , , and are positive real parameters, is a function satisfying suitable regularity and growth conditions, and is the fractional Laplace operator defined as

Fractional Laplace operators have been proved to be valuable tools in the modeling of many phenomena in various fields, such as minimal surfaces, quasi-geostrophic flows, conservation laws, optimization, multiple scattering, anomalous diffusion, ultrarelativistic limits of quantum mechanics, finance, phase transitions, stratified materials, crystal dislocation, semipermeable membranes, flame propagation, soft thin films, and materials science. Recently, there has been significant development in fractional Laplace operators; for examples, see [113] and the references therein.

Motivated and inspired by the papers [1315], in this paper, a variational approach is provided to investigate the existence of three weak solutions to a perturbed nonlocal fractional Laplacian equation (2), by using a three-critical-point theorem obtained by Bonanno and Marano in [14].

2. Preliminaries

Let such that , . The classical fractional Sobolev space is defined by endowed with the norm (the so-called Gagliardo norm) Let By [6] in the sequel we can take the function as norm on , where . It is easily seen that is a Hilbert space, with scalar product Since , we have that the integral in (7) (and in the related scalar product) can be extended to all .

By a weak solution of (2) we mean a function such that for all .

Denote by the first eigenvalue of the operator with homogeneous Dirichlet boundary data

For the existence and the basic properties of we may refer to [7]. From [7, 16], we know that if then we can take a norm on as follows: Moreover, we have where

Remark 1. If , then
Taking into account Lemma 8 in [6], we know that the embedding is continuous for any , while it is compact whenever . Thus, form any there exists a positive constant such that for any .
Let ; simple calculations show that there is such that .
Set

Lemma 2. Let , , , and let be defined by (16). Then , and there exist and such that where is as in (14).

Proof. By Proposition 3.4 in [5], we have where Here . From the trivial inequality , , and (18), we obtain Moreover, according to the definition of norm for , we get By [5], we have where By polar coordinates, for any , we obtain where is the Lebesgue measure of the unit sphere in . Furthermore, we have Thanks to (20)–(25), we conclude that which implies that . By (12), (14), and (26), we obtain where Hence, the conclusion of right-hand side of (17) holds.
On the other hand, we have For and , we have Thus, where . Moreover, we obtain where Substitute (31) and (32) into (29), we get From (34) and (12), we obtain where Thus, the conclusion of left-hand side of (17) holds.

In this paper our main tool is a three-critical-point theorem of [14] which is recalled below.

Theorem 3 (see [14]). Let be a reflexive real Banach space; let be a coercive, continuously Gâteaux differentiable, and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on , and let be a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact such that Assume that there exist and , with , such that(i) ;(ii)for each the functional is coercive.Then, for each , the functional has at least three distinct critical points in .

3. Main Result

Let be a Carathéodory function such that(H1)there exist and , , such that

Theorem 4. Let function satisfy condition (H1). Assume that (H2) for all ;(H3)there exist two positive constants and such that for almost every and for every ;(H4)let such that there exist two positive constants and , with such that where and two positive constants and are as in (28) and (36), respectively. Then, for every belonging to problem (2) possesses at least three weak solutions in .

Proof. Let us apply Theorem 3 with and where For each , one has
From the proof of Theorem 1 in [16], we obtain that is coercive, continuously Gâteaux differentiable, and sequentially weakly lower semicontinuous functional. Moreover, similar to the proof of proposition in [17], we get by (12) that for every and belonging to . This actually means that is a uniformly monotone operator in . In addition, standard arguments ensure that also turns out to be coercive and hemicontinuos in . Therefore, admits that a continuous inverse in follows immediately by applying Theorem 26. A. of [18]. Furthermore, the functional is well defined, continuously Gâteaux differentiable with compact derivative and .
By [16] we know that being a weak solution of problem (2) is equivalent to being a critical point of the functional . Since , from Lemma 2, one has Bearing in mind that (H4), it follows that . By (H2), we obtain By Lemma 2, we have So, by (48) and (49), one has Thanks to (H1), one has Thus, by (15) and (51), for every , we obtain Therefore
Denote the function By (53), we have
Owing to (50), (55), and (H4), we have Hence, the assumption (i) of Theorem 3 is satisfied.
Furthermore, if , for each , , Hölder's inequality and (15) give Due to (H3) and (57), we deduce that Hence, is a coercive functional for every positive parameter , in particular, for each . So also condition (ii) holds. So all the assumptions of Theorem 3 are satisfied. Thus, for each there exists , depending on , such that, for any , the functional has at least three distinct critical points that are weak solutions to problem (2).

Remark 5. Similar to Example 3.1 in [15], we can give a concrete example of function satisfying hypotheses (H1)–(H4). Set , , and and let where From (H4) we know that and . Let be a positive constant such that and consider the following continuous and positive function : Obviously, for each , and (H1) holds. Furthermore, for every , we have Thus the conditions (H2) and (H3) are satisfied. Moreover, and which implies that (H4) holds.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors thank the referees for their valuable and helpful suggestions and comments that improved the paper. This work is supported by Natural Science Foundation of Jiangsu Province (BK2011407) and Natural Science Foundation of China (11271364 and 10771212).

References

  1. L. A. Caffarelli, S. Salsa, and L. Silvestre, “Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian,” Inventiones Mathematicae, vol. 171, no. 2, pp. 425–461, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. L. Silvestre, “Regularity of the obstacle problem for a fractional power of the Laplace operator,” Communications on Pure and Applied Mathematics, vol. 60, no. 1, pp. 67–112, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. X. Cabré and J. Tan, “Positive solutions of nonlinear problems involving the square root of the Laplacian,” Advances in Mathematics, vol. 224, no. 5, pp. 2052–2093, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. X. Cabré and Y. Sire, “Nonlinear equations for fractional Laplacians I: regularity, maximum principles, and Hamiltonian estimates,” Annales de l'Institut Henri Poincare C, 2013. View at Publisher · View at Google Scholar
  5. 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 Zentralblatt MATH · View at MathSciNet
  6. R. Servadei and E. Valdinoci, “Mountain pass solutions for non-local elliptic operators,” Journal of Mathematical Analysis and Applications, vol. 389, no. 2, pp. 887–898, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. R. Servadei and E. Valdinoci, “Variational methods for non-local operators of elliptic type,” Discrete and Continuous Dynamical Systems A, vol. 33, no. 5, pp. 2105–2137, 2013. View at Zentralblatt MATH · View at MathSciNet
  8. R. Servadei and E. Valdinoci, “Lewy-Stampacchia type estimates for variational inequalities driven by (non)local operators,” Revista Matemática Iberoamericana, vol. 29, no. 3, pp. 1091–1126, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. M. M. Fall and T. Weth, “Nonexistence results for a class of fractional elliptic boundary value problems,” Journal of Functional Analysis, vol. 263, no. 8, pp. 2205–2227, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. F. Ferrari and I. E. Verbitsky, “Radial fractional Laplace operators and Hessian inequalities,” Journal of Differential Equations, vol. 253, no. 1, pp. 244–272, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. R. Servadei and E. Valdinoci, “The Brezis-Nirenberg result for the fractional Laplacian,” Transactions of the American Mathematical Society. In press. View at Publisher · View at Google Scholar
  12. C. Bai, “Existence results for non-local operators of elliptic type,” Nonlinear Analysis A, vol. 83, pp. 82–90, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. S. Dipierro, G. Palatucci, and E. Valdinoci, “Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian,” Le Matematiche, vol. 68, no. 1, pp. 201–216, 2013. View at MathSciNet
  14. G. Bonanno and S. A. Marano, “On the structure of the critical set of non-differentiable functions with a weak compactness condition,” Applicable Analysis, vol. 89, no. 1, pp. 1–10, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. G. Bonanno and G. Molica Bisci, “Three weak solutions for elliptic Dirichlet problems,” Journal of Mathematical Analysis and Applications, vol. 382, no. 1, pp. 1–8, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. G. Molica Bisci and R. Servadei, “A bifurcation result for non-local fractional equations,” Journal of Applied Analysis. In press. View at Publisher · View at Google Scholar
  17. G. Bonanno and P. Candito, “Three solutions to a Neumann problem for elliptic equations involving the p-Laplacian,” Archiv der Mathematik, vol. 80, no. 4, pp. 424–429, 2003. View at MathSciNet
  18. E. Zeidler, Nonlinear Functional Analysis and Its Applications, vol. 2, Springer, New York, NY, USA, 1985.