Infinitely many solutions for a nonlinear Schrödinger equation with non-symmetric electromagnetic fields

Abstract

In this paper, we study the nonlinear Schrödinger equation with non-symmetric electromagnetic fields $$(\frac{\nabla }{i}-A_{\epsilon }x){)}^{2}u+V_{\epsilon }(x)u=f(u),u\in H^1({\mathbb{R}}^N,\mathbb{C}),$$ where $A_{\epsilon }(x)=(A_{\epsilon ,1}(x),A_{\epsilon ,2}(x),\cdots ,A_{\epsilon ,N}(x))$ is a magnetic field satisfying that $A_{\epsilon ,j}(x)(j=1,\dots ,N)$ 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.

arXiv comment39fages, 0 figures. arXiv admin note: text overlap with arXiv:1210.8209, arXiv:1209.2824 by other authors