Groundstates and infinitely many solutions for the Schrödinger-Poisson equation with magnetic field

Abstract

<p style='text-indent:20px;'>In this paper, we investigate the nonlinear Schrödinger-Poisson equation with magnetic field. By combining non-Nehari manifold method and some new energy estimate inequalities, we obtain the existence of a ground state solution, where the strict monotonicity condition and the Ambrosetti-Rabinowitz growth condition are not needed. Moreover, when both the potential and the nonlinearity are sign-changing, by applying the Fountain Theorem and some analytical techniques, we prove the existence of infinitely many solutions. Our results extend and improve the present ones in the literature.</p>