Multiple complex-valued solutions for the nonlinear Schrödinger equations involving magnetic potentials

Gan Lu; Weiming Liu

doi:10.3934/cpaa.2017096

Article Contents

2017, Volume 16, Issue 6: 1957-1975. Doi: 10.3934/cpaa.2017096

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Multiple complex-valued solutions for the nonlinear Schrödinger equations involving magnetic potentials

Gan Lu and
Weiming Liu^,

School of Mathematics and Statistics, Hubei Normal University, Huangshi, 435002, China

* Corresponding author
* Corresponding author

Received: August 31, 2016

Revised: March 31, 2017

Published: September 2017

The authors would like to thank the referees for their suggestions of this work. The paper is supported by the fund from NSFC (No.11601139)

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Abstract

This paper is concerned with the following nonlinear Schrödinger equations with magnetic potentials

$\begin{equation}\label{0} \Bigl(\frac{\nabla}{i}-α A(|x|)\Bigl)^{2}u+(1+α V(|x|))u=|u|^{p-2}u,\,\,u∈ H^{1}(\mathbb{R}^{N},\mathbb{C}),\ \ \ \ \ \ \ \ \ \ \left( 0.1 \right) \end{equation}$

where $2<p<\frac{2N}{N-2}$ if $N≥q 3$ and $2<p<+∞$ if $N=2$. $α$ can be regarded as a parameter. $A(|x|)=(A_{1}(|x|),A_{2}(|x|),···,A_{N}(|x|))$ is a magnetic field satisfying that $A_{j}(|x|)>0(j=1,...,N)$ is a real $C^{1}$ bounded function on $\mathbb{R}^{N}$ and $V(|x|)>0$ is a real continuous electric potential. Under some decaying conditions of both electric and magnetic potentials which are given in section 1, we prove that the equation has multiple complex-valued solutions by applying the finite reduction method.

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Mathematics Subject Classification: Primary: 35J10, 35B99; Secondary: 35J60.

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