Energy solutions and concentration problem of fractional Schr\xf6dinger equation

Peiluan Li,Yuan Yuan

doi:10.1186/s13661-018-0940-z

Abstract

In this paper, we consider a fractional Schrödinger equation with steep potential well and sublinear perturbation. By virtue of variational methods, the existence criteria of infinitely many nontrivial high or small energy solutions are established. In addition, the phenomenon of the concentration of solutions is also explored. We also give some examples to demonstrate the main results.

Highlights

1 Introduction In this paper, we are concerned with the following fractional Schrödinger equations: (– )αu + ρV (x)u = f (x, u) + h(x)|u|p–2u, x ∈ RN, (1.1)
The variational method has been used in many works to study the fractional Schrödinger equations (1.2)
2 Preliminaries we review some definitions and related lemmas

Summary

Introduction

We are concerned with the following fractional Schrödinger equations:. The existence and multiplicity of nontrivial solutions for the integer order Schrödinger equation have been extensively investigated. For the different cases of the potential V and the nonlinearity f , some researchers have investigated the fractional Schrödinger equations under the appropriate assumptions:. The variational method has been used in many works to study the fractional Schrödinger equations (1.2). In [20], when V = 1 and f (x, u) = f (u), the authors gave the existence of least two nontrivial radial solutions without the A–R condition by variational methods. In [22], the authors studied the existence of infinitely many nontrivial energy solutions by variational methods. By variational methods, we have established existence criteria of infinitely many nontrivial high or small energy solutions without A–R condition.

There exist two constants σ

RN RN

From the condition h

Let βk