Infinitely many weak solutions for a fractional Schrödinger equation

Wei Dong,Jiafa Xu,Zhongli Wei

doi:10.1186/s13661-014-0159-6

Abstract

In this paper we are concerned with the fractional Schrödinger equation ( − Δ ) α u+V(x)u=f(x,u), x∈ R N , where 0<α<1, N>2α, ( − Δ ) α stands for the fractional Laplacian of order α, V is a positive continuous potential, and f is a continuous subcritical nonlinearity. We obtain the existence of infinitely many weak solutions for the above problem by the fountain theorem in critical point theory.

Highlights

In this paper we consider the following fractional Schrödinger equation:(– )αu + V (x)u = f (x, u), x ∈ RN, ( . )where < α α, (– )α stands for the fractional Laplacian of order α, and the potential V : RN → R is a continuous function satisfying (V) < infx∈RN V (x) = V < lim inf|x|→∞ V (x) = V∞ < ∞
Where < α α, (– )α stands for the fractional Laplacian of order α, and the potential V : RN → R is a continuous function satisfying (V) < infx∈RN V (x) = V < lim inf|x|→∞ V (x) = V∞ < ∞
In [ ], Shang et al considered the existence of nontrivial solutions for ( . ) with f (u) = |u|q– u, where < q < ∗α

Summary

Introduction

In this paper we consider the following fractional Schrödinger equation:. where < α < , N > α, (– )α stands for the fractional Laplacian of order α, and the potential V : RN → R is a continuous function satisfying (V) < infx∈RN V (x) = V < lim inf|x|→∞ V (x) = V∞ < ∞. In this paper we consider the following fractional Schrödinger equation:. In [ ], Shang and Zhang considered the critical fractional Schrödinger equation ε α(– )αu + V (x)u = |u| ∗α– u + λf (u), x ∈ RN , where ε and λ are positive parameters, V and f satisfy (V) and (H ), respectively. We can see that an alternative definition of the fractional Sobolev space Hα(RN ) via the Fourier transform is as follows: Hα RN = u ∈ L RN : + |ξ | α |F u| dξ < +∞.

Xj and

Then there exists k such that ρk

Now for any k

Thus we obtain