On the Schrödinger equation involving a critical Sobolev exponent and magnetic field

Abstract

We consider the semilinear Schrödinger equation $$ -\Delta_A u+V(x)u =Q(x)|u|^{2^{*}-2}u. $$ Assuming that $V$ changes sign, we establish the existence of a solution $u\ne 0$ in the Sobolev space $ H_{A,V^+}^{1}(\mathbb R^N)$. The solution is obtained by a min-max type argument based on a topological linking. We also establish certain regularity properties of solutions for a rather general class of equations involving the operator $-\Delta_A$.