Ground-State Solutions for Asymptotically Cubic Schrödinger–Maxwell Equations

Abstract

In this paper, using variational methods and critical point theory, we study the existence of ground-state solutions for the following nonlinear Schrodinger–Maxwell equations $$\left\{\begin{array}{l@{\quad}l} -\triangle u + V(x)u + \phi u = f(x, u), & {\rm in}\, \mathbb{R}^{3},\\ -\triangle\phi = 4\pi u^{2}, & {\rm in} \, \mathbb{R}^{3},\end{array}\right. $$ (NSM) where f is asymptotically cubic, V 1-periodic in each of \({x_1, x_2, x_3}\) and \({\underline{V}:= {\rm inf}_{x\in\mathbb{R}^3}V(x) > 0}\). Under some more assumptions on V and f, we develop a direct and simple method to find ground-state solutions for \({(\mathrm{NSM})}\). The main idea is to find a minimizing (PS) sequence for the energy functional outside the Nehari manifold \({\mathcal{N}}\) using the diagonal method. This seems to be the first result for \({(\mathrm{NSM})}\) satisfying the assumptions (V) and (N).