Multiple solutions to nonlinear Schrödinger equations with singular electromagnetic potential

Abstract

We consider the semilinear electromagnetic Schrödinger equation (-i\nabla+A(x))^{2}u + V(x)u = |u|^{2^{\ast}-2}u, u\in D_{A,0}^{1,2}(Ω,\mathbb{C}), where $\Omega =({\mathbb{R}}^m\setminus 0)\times {\mathbb{R}}^{N-m}$ with $2\le m\le N$, $N\ge 3$, $2^{\ast }:=2N/(N-2)$ 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.

Journal of Fixed Point Theory and Applications, to appear