Dynamics of Partial Differential Equations

Volume 14 (2017)

Number 2

Multiple nontrivial solutions for a class of nonlinear Schrödinger equations with linear coupling

Pages: 159 – 200

DOI: https://dx.doi.org/10.4310/DPDE.2017.v14.n2.a3

Authors

Jun Wang (Faculty of Science, Jiangsu University, Zhenjiang, Jiangsu, China)

Zhaosheng Feng (Department of Mathematics, University of Texas Rio Grande Valley, Edinburg, Tx., U.S.A.)

Abstract

In this paper, we are concerned with the existence and multiplicity of nontrivial solutions of a class of nonlinear Schrödinger equations which arise from nonlinear optics. We prove that there are two families of semiclassical positive solutions, which concentrate on the minimal and maximum points of the associated potentials, respectively. We also investigate the relationship between the number of solutions and the topology of the set of the global minima of the potentials by the minimax theorem. The novelty is that it might be the first attempt to explore multiplicity and concentration of positive solutions for such kind of coupled Schrödinger equations.

Keywords

variational methods, Schrödinger equations, positive solutions, ground state solutions, Nehari manifolds, category theory

2010 Mathematics Subject Classification

Primary 35B09, 35J60. Secondary 35J20, 35M10.

Published 31 May 2017