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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 809769, 7 pages
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 ####^~^~^~^~^~^####xa9; 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.
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.
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 [1####^~^~^~^~^~^####x2013;13] and the references therein.
Motivated and inspired by the papers [13####^~^~^~^~^~^####x2013;15], 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 .
Let such that , . The classical fractional Sobolev space is defined by endowed with the norm (the so-called Gagliardo norm) Let By  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
Remark 1. If , then
Taking into account Lemma 8 in , 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 .
Proof. By Proposition 3.4 in , we have
Here . From the trivial inequality , , and (18), we obtain
Moreover, according to the definition of norm for , we get
By , we have
By polar coordinates, for any , we obtain
where is the Lebesgue measure of the unit sphere in . Furthermore, we have
Thanks to (20)####^~^~^~^~^~^####x2013;(25), we conclude that
which implies that . By (12), (14), and (26), we obtain
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  which is recalled below.
Theorem 3 (see ). Let be a reflexive real Banach space; let be a coercive, continuously G####^~^~^~^~^~^####xe2;teaux differentiable, and sequentially weakly lower semicontinuous functional whose G####^~^~^~^~^~^####xe2;teaux derivative admits a continuous inverse on , and let be a continuously G####^~^~^~^~^~^####xe2;teaux differentiable functional whose G####^~^~^~^~^~^####xe2;teaux 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 Carath####^~^~^~^~^~^####xe9;odory 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
For each , one has
From the proof of Theorem 1 in , we obtain that is coercive, continuously G####^~^~^~^~^~^####xe2;teaux differentiable, and sequentially weakly lower semicontinuous functional. Moreover, similar to the proof of proposition in , 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 . Furthermore, the functional is well defined, continuously G####^~^~^~^~^~^####xe2;teaux differentiable with compact derivative and .
By  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 , , H####^~^~^~^~^~^####xf6;lder'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 , we can give a concrete example of function satisfying hypotheses (H1)####^~^~^~^~^~^####x2013;(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.
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).
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