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

In this paper, we study the nonlinear Schr"{o}dinger equation with non-symmetric electromagnetic fields $$(\frac{\mathrm{\nabla }}{i}-A_{ϵ}x){)}^{2}u+V_{ϵ}(x)u=f(u),u\in H^1({\mathbb{R}}^N,\mathbb{C}),$$ where $A_{ϵ}(x)=(A_{ϵ,1}(x),A_{ϵ,2}(x),\cdots ,A_{ϵ,N}(x))$ is a magnetic field satisfying that $A_{ϵ,j}(x)(j=1,\dots ,N)$ is a real $C^1$ bounded function on ${\mathbb{R}}^N$ and $V_{ϵ}(x)$ 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 ${ϵ}_0>0$ such that for $0 < \epsilon < \epsilon_{0 }$, the above problem has infinitely many complex-valued solutions.