Existence of Solutions for a Fractional Laplacian Equation with Critical Nonlinearity

and Applied Analysis 3 u = 0. There is a great deal of work on μ 1 (Ω); see for example [9]. We have

The study of existence and concentration of the semiclassical states of Schrödinger equation goes back to the pioneer work [3] by Floer and Weinstein.Ever since then, equations of (3) type with subcritical nonlinearities ( < 2 * = 2/( − 2) for  ≥ 3) have been studied by many authors.For critical nonlinearity ( = 2 * for  ≥ 4), Clapp and Ding [1,2] established the existence and multiplicity of positive solutions and minimal nodal solutions which localize near the potential well for  small and  large.
The fractional Schrödinger equation is a fundamental equation of fractional quantum mechanics.It was discovered by Nick Laskin as a result of extending the Feynman path integral, from the Brownian-like to Lévy-like quantum mechanical paths.The term fractional Schrödinger equation was coined by Nick Laskin.
Recently, a great attention has been devoted to the fractional and nonlocal operators of elliptic type, both for their interesting theoretical structure and in view of concrete applications in many fields such as combustion and dislocations in mechanical systems.This type of operator seems to have a prevalent role in physical situations and has been studied by many authors [4][5][6][7][8][9] and references therein.In [5], Di Nezza et al. deal with the fractional Sobolev space  , and analyze their role in the trace theory.They prove continuous and compact embeddings, investigating the problem of the extension domains and other regularity results.In [8], Felmer et al. proved the existence of positive solutions of nonlinear Schrödinger equation involving the fractional Laplacian in R  .They further analyzed regularity, decay, and symmetry properties of these solutions.Servadei and Valdinoci [9] studied the existence of nontrivial solutions for equations driven by a nonlocal integrodifferential operator   with homogeneous Dirichlet boundary conditions.They give more general and more precise results about the eigenvalues of a linear operator.
The fractional Laplace operator (−Δ)  in (4) can be defined as We say that a function  ∈   (R  ) solves (4) in the weak sense if Define the energy functional by Then we know the critical points of   are exactly the weak solutions of (7).In this sense we will prove the existence of the critical points of the functional   .Fréchet derivative of   is Concerning the Schrödinger equation: Clapp and Ding [1] proved the following.
Here   is defined as where  is an  2 (R  ) space with potential and will be defined in Section 2.
This paper is organized as follows.In Section 2, we give some preliminary results.In Section 3, we finish the proof of Theorem 1.In Section 4, we finish the proof of Theorem 2.
We consider the fractional Sobolev space: with norm And let be the Hilbert space equipped with norm If  > 0, then it is equivalent to the norms Thus  is continuously embedded in   (R  ).
Remark 3. We know the embedding   (R  ) →  ] (R  ) is continuous; see [5] or [8].So the embedding Thanks to Remark 3, we can define the constant   as in formula ( 14) and get that   > 0.

The Proof of Theorem 1
In this section we will finish the proof of Theorem 1.
The critical points of   lie on the Nehari manifold Since 0 <  <  1 (Ω) and 2 < 2 * (), the function  ∈ R + →   () has a unique maximum point () > 0 and () ∈ .Define and we observe that From Lemma 5, the constant  1 is positive.On the other hand, we define where So, We consider the functional on   0 (Ω).Its Nehari manifold where   is defined in formula (14) and   is given in Lemma 5.
Hence, Proposition 7 is proved.Proof.We proceed by steps. Step Thus {  } is bounded in .

Proof. By
Step 1 and  is a reflexive space, up to a subsequence, still denoted by   , there exists (56) Passing to the limit in this expression as  → +∞ and taking into account (51), (53), and (55), we get for any  ∈ ; that is,  ∞ is a solution of problem (7).
Step 3. The following equality holds true: Proof.By Step 2, taking  =  ∞ ∈  as a test function in (7), we have So we get Hence, Step 3 is proved.Now, we conclude the proof of Proposition 8.
We have finished the proof of Theorem 1 by Proposition 8.
By definition of  1 and Proposition 6, there exists a minimizing sequence for   on , and we note {  }.By Ekeland's variational principle, we may assume that it is a Palais Smale sequence.So we have   (  ) → ,   has at least one critical point with critical value  for each 0 <  <  1 (Ω) and  ≥ ().
1.The sequence {  } is bounded in .