Multiplicity of solutions of perturbed Schrödinger equation with electromagnetic fields and critical nonlinearity in R N

Sihua Liang,Yanli Song

doi:10.1186/s13661-014-0240-1

Abstract

In this paper, we deal with the existence and multiplicity of solutions for perturbed Schrödinger equation with electromagnetic fields and critical nonlinearity in : for all , where , is a nonnegative potential. By using Lions’ second concentration compactness principle and concentration compactness principle at infinity to prove that the condition holds locally and by variational method, we show that this equation has at least one solution provided that , for any , it has m pairs of solutions if , where ℰ and are sufficiently small positive numbers. MSC: 35J60, 35B33.

Highlights

1 Introduction In this paper, we are concerned with the existence of nontrivial solutions to the following perturbed Schrödinger equation with electromagnetic fields and critical nonlinearity in RN :
This paper is motivated by some works concerning the nonlinear Schrödinger equation i
The main purpose of this paper is to study the existence and multiplicity of solutions of perturbed Schrödinger equations with electromagnetic fields and critical nonlinearity ( . )

Summary

Introduction

We are concerned with the existence of nontrivial solutions to the following perturbed Schrödinger equation with electromagnetic fields and critical nonlinearity in RN :. The main purpose of this paper is to study the existence and multiplicity of solutions of perturbed Schrödinger equations with electromagnetic fields and critical nonlinearity In accordance with the Sobolev critical nonlinearities, there have been many papers devoted to studying the existence of solutions to elliptic boundary-valued problems on bounded domains after the pioneering work by Brezis and Nirenberg [ ]. Ding and Lin [ ] first studied the existence of semi-classical solutions to the problem on the whole space with critical nonlinearities and established the existence of positive solutions as well as of those that change sign exactly once.

Consider the functional

It is easy to prove that

Jλ cifλi

In virtue of know that