Existence of solution for magnetic Schrödinger equation with the Neumann boundary condition

Abstract

We consider the nonlinear Schrödinger equation with magnetic field and the Neumann boundary condition: where Ω is a boundary domain in with a boundary, ν is the outward normal vector field at , , , ( is given by (3)), is a magnetic vector potential. When the exponent is subcritical, we can obtain solutions by Nehari manifold. When the exponent is critical, , we can obtain solutions by constrained minimization arguments.