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

TL;DR

This paper investigates the existence of multiple solutions to a nonlinear Schr"odinger equation with singular electromagnetic potentials, including Aharonov-Bohm potentials, under symmetry conditions.

Contribution

It establishes the multiplicity of solutions for a critical nonlinear Schr"odinger equation with singular magnetic and Hardy potentials, extending previous results to include symmetry considerations.

Findings

Multiple solutions exist under symmetry assumptions.

Solutions exhibit specific symmetry properties.

The results include cases with Aharonov-Bohm potentials.

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.