Schrodinger-Maxwell systems on compact Riemannian manifolds

Abstract

In this paper we are focusing to the following Schrödinger-Maxwell system $({\mathcal{SM}}_{\Psi (\lambda ,\cdot )}^e)$: \[ \begin{cases} -Δ_{g}u+β(x)u+euϕ=Ψ(λ,x)f(u) & \mathrm{in}\ M -Δ_{g}ϕ+ϕ=qu^{2} & \mathrm{\mathrm{in}\ M} \end{cases} \] 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.

arXiv comment15 pages