Nontrivial solutions for a mixed boundary problem for Schrödinger equations with an external magnetic field

Abstract

We study the existence of solutions for a class of nonlinear Schr\"{o}dinger equations involving a magnetic field with mixed Dirichlet--Neumann boundary conditions. We use Lusternik--Shnirelman category and the Morse theory to estimate the number of nontrivial solutions in terms of the topology of the part of the boundary where the Neumann condition is prescribed.