Schrödinger–Maxwell systems on compact Riemannian manifolds

Abstract

In this paper, we are focusing to the following Schrödinger–Maxwell system: \begin{equation} \begin{cases} -\Delta_{g}u+\beta(x)u+eu\phi=\Psi(\lambda,x)f(u) & \mbox{in} \ M,\\ -\Delta_{g}\phi+\phi=qu^{2} & \mbox{in} \ M, \end{cases}\tag{$\mathcal{SM}_{\Psi(\lambda,\cdot)}^{e}$} \end{equation} where $(M,g)$ is a 3-dimensional compact Riemannian manifold without boundary, $e,q>0$ are positive numbers, $f:\mathbb{R}\to\mathbb{R}$ is a continuous function, $\beta\in C^{\infty}(M)$ and $\Psi\in C^{\infty}(\mathbb{R}_{+}\times M)$ are positive functions. By various variational approaches, existence of multiple solutions of the problem $(\mathcal{SM}_{\Psi(\lambda,\cdot)}^{e})$ is established.

Keywords

• schrödinger–maxwell systems
• critical points
• compact riemannian manifolds