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

In this paper, we study the nonlinear Schrödinger equation with non-symmetric electromagnetic fields $\Big(\frac{\nabla}{i}-A_ε x)\Big)^2 u+V_ε(x)u=f(u),\ u\in H^1 (\mathbb{R}^N,\mathbb{C}),$ where $A_ε(x)=(A_{ε,1}(x),A_{ε,2}(x),\cdots,A_{ε,N}(x))$ is a magnetic field satisfying that $A_{ε,j}(x)(j=1,\ldots,N)$ 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 $0 < ε< ε_{0 }$, the above problem has infinitely many complex-valued solutions.

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