On the Existence of Solutions for the Critical Fractional Laplacian Equation inℝN

Zifei Shen,Fashun Gao

doi:10.1155/2014/143741

Abstract

We study existence of solutions for the fractional Laplacian equation-Δsu+Vxu=u2*s-2u+fx, uinℝN,u∈Hs(RN), with critical exponent2*s=2N/(N-2s),N>2s,s∈0, 1, whereVx≥0has a potential well andf:ℝN×ℝ→ℝis a lower order perturbation of the critical poweru2*s-2u. By employing the variational method, we prove the existence of nontrivial solutions for the equation.

Highlights

In the last 20 years, the classical nonlinear Schrodinger equation has been extensively studied by many authors [1,2,3,4,5,6,7,8,9,10] and the references therein
A great attention has been focused on the study of problems involving the fractional Laplacian recently. This type of operator seems to have a prevalent role in physical situations such as combustion and dislocations in mechanical systems or in crystals
The aim of this paper is to find solutions for (5) by variational methods

Summary

Introduction

In the last 20 years, the classical nonlinear Schrodinger equation has been extensively studied by many authors [1,2,3,4,5,6,7,8,9,10] and the references therein. In [3], Rabinowitz proved the existence of standing wave solutions of nonlinear Schrodinger equations. A great attention has been focused on the study of problems involving the fractional Laplacian recently This type of operator seems to have a prevalent role in physical situations such as combustion and dislocations in mechanical systems or in crystals. There exists θ > 0 such that K(x) ≥ θ|x|−(N+2s) and K(x) = K(−x) for any x ∈ RN \ {0} They proved that problem (1) admits a nontrivial solution for any λ > 0. They studied the case f(x, u) ≡ 0 and K(x) = |x|−(N+2s), respectively.

Preliminary Results

The Proof of Theorem 1

The Proof of Theorem 2