Infinitely many solutions for a nonlinear Schr\"{o}dinger equation with non-symmetric electromagnetic fields

TL;DR

This paper proves the existence of infinitely many solutions for a nonlinear Schrödinger equation with non-symmetric electromagnetic fields, using localized energy methods under certain decay and nondegeneracy conditions.

Contribution

It establishes the existence of infinitely many solutions for a class of nonlinear Schrödinger equations with non-symmetric electromagnetic fields, extending previous results to more general magnetic and electric potentials.

Findings

Existence of infinitely many solutions for small epsilon

Solutions are complex-valued and constructed via localized energy methods

Results hold under decay and nondegeneracy conditions

Abstract

In this paper, we study the nonlinear Schr\"{o}dinger equation with non-symmetric electromagnetic fields $$\Big(\frac{\nabla}{i}-A_{\epsilon} x)\Big)^2 u+V_{\epsilon}(x)u=f(u),\ u\in H^1 (\mathbb{R}^N,\mathbb{C}), $$ where $A_{\epsilon}(x)=(A_{\epsilon,1}(x),A_{\epsilon,2}(x),\cdots,A_{\epsilon,N}(x))$ is a magnetic field satisfying that $A_{\epsilon,j}(x)(j=1,\ldots,N)$ is a real $C^{1}$ bounded function on $\mathbb{R}^{N}$ and $V_{\epsilon}(x)$ is an electric potential. Both of them satisfy some decay conditions and $f(u)$ is a nonlinearity satisfying some nondegeneracy condition. Applying localized energy method, we prove that there exists some $\epsilon_{0 }> 0$ such that for $0 < \epsilon < \epsilon_{0 }$, the above problem has infinitely many complex-valued solutions.