Multiple solutions for a NLS equation with critical growth and magnetic field

Abstract

In this paper, we are concerned with the multiplicity of nontrivial solutions for the following class of complex problems $$ (-i\nabla - A(μx))^{2}u= μ|u|^{q-2}u + |u|^{2^{*}-2}u \ \mbox{in} \ Ω, \ \ \ \ u \in H^{1}_{0}(Ω,\mathbb{C}), $$ where $\Omega \subset {\mathbb{R}}^N(N\ge 4)$ is a bounded domain with smooth boundary. Using the Lusternik-Schnirelman theory, we relate the number of solutions with the topology of $\Omega$.