Three Weak Solutions for Nonlocal Fractional Laplacian Equations

and Applied Analysis 3 Moreover, according to the definition of norm forH(R), we get 󵄩󵄩󵄩uδ 󵄩󵄩󵄩 2 H 1 (Rn) = ∫ Rn 󵄨󵄨󵄨∇uδ (x) 󵄨󵄨󵄨 2 dx


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  ∈ (0, 1) is fixed, Ω is a nonempty bounded open subset of R  ,  > 2,  and  are positive real parameters,  : Ω × R → R is a function satisfying suitable regularity and growth conditions, and (−Δ)   (3) 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][2][3][4][5][6][7][8][9][10][11][12][13] and the references therein.
Motivated and inspired by the papers [13][14][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-criticalpoint theorem obtained by Bonanno and Marano in [14].

Preliminaries
Let  ∈ (0, 1) such that 2 < , Ω ⊂ R  .The classical fractional Sobolev space   (R  ) 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  0 , where O = (CΩ)×(CΩ) ⊂ R  ×R  .It is easily seen that ( 0 , ‖ ⋅ ‖  0 ) is a Hilbert space, with scalar product Since V ∈  0 , we have that the integral in (7) (and in the related scalar product) can be extended to all R  × R  .By a weak solution  of ( 2 for all  ∈  0 .Denote by  1 > 0 the first eigenvalue of the operator (−Δ)  with homogeneous Dirichlet boundary data For the existence and the basic properties of  1 we may refer to [7].From [7,16], we know that if  <  1 then we can take a norm on  0 as follows: Moreover, we have where Remark 1.If 0 <  <  1 , then Taking into account Lemma 8 in [6], we know that the embedding  :  0 →  ] (R  ) is continuous for any ] ∈ [1, 2 * ], while it is compact whenever ] ∈ [1, 2 * ).Thus, form any ] ∈ [1, 2 * ) there exists a positive constant  ] such that for any V ∈  0 .
Then, for each  ∈ Λ  , the functional Φ − Ψ has at least three distinct critical points in .

Main Result
Let  : Ω × R → R be a Carathéodory function such that (H1) there exist  1 ,  2 ≥ 0 and  ∈ (1, 2 for almost every  ∈ Ω and for every V ∈ R; (H4) let 0 <  <  1 such that there exist two positive constants  and , with  > √2/ * (/  ) such that where and two positive constants  * and  * are as in (28) and (36), respectively.Then, for every  belonging to For each , V ∈  0 , one has From the proof of Theorem 1 in [16], we obtain that Φ  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 V belonging to  0 .This actually means that Φ   is a uniformly monotone operator in  0 .In addition, standard arguments ensure that Φ   also turns out to be coercive and hemicontinuos in  0 .Therefore, Φ   admits that a continuous inverse in  * 0 follows immediately by applying Theorem 26. A. of [18].Furthermore, the functional Ψ is well defined, continuously Gâteaux differentiable with compact derivative and Φ  (0) = Ψ(0) = 0.