Multiple solutions to nonlinear Schr\"odinger equations with singular electromagnetic potential

Abstract

We consider the semilinear electromagnetic Schr\"{o}dinger equation (-i\nabla+A(x))^{2}u + V(x)u = |u|^{2^{\ast}-2}u, u\in D_{A,0}^{1,2}(\Omega,\mathbb{C}), where $\Omega=(\mathbb{R}^{m}\smallsetminus{0})\times\mathbb{R}^{N-m}$ with $2\leq m\leq N$, $N\geq3$, $2^{\ast}:= 2N/(N-2)$ is the critical Sobolev exponent, $V$ is a Hardy term and $A$ is a singular magnetic potential of a particular form which includes the Aharonov-Bohm potentials. Under some symmetry assumptions on $A$ we obtain multiplicity of solutions satisfying certain symmetry properties.