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In this paper, we study the nonlinear Schr\"{o}dinger equation withnon-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 anelectric potential. Both of them satisfy some decay conditions and $f(u)$ is anonlinearity satisfying some nondegeneracy condition. Applying localized energymethod, we prove that there exists some $\epsilon_{0 }> 0$ such that for $0 <\epsilon < \epsilon_{0 }$, the above problem has infinitely many complex-valuedsolutions.

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