Abstract:

In this paper, we study the existence of the ground state solutions for the following Schrödinger-Maxwell equations

$\left\{\begin{array}{ll} -\triangle u+V(x)u-(K(x)+\alpha)\phi u=\beta|u|^{4}u+b(x)|u|^{p-1}u, &(x,u)\in(\mathbb{R}^{3},\mathbb{R}),\\ \triangle\phi=(K(x)+\alpha)u^{2},& (x,u)\in(\mathbb{R}^{3},\mathbb{R}) \end{array} \right.$

where β is a positive constant. Under some assumptions on V, K and b(x), by using the variational method and critical point theorem, we prove that such a class of equations has at least a ground state solution for α < 0 and p ∈ (3, 4).

Key words: Schrödinger-Maxwell equations, Critical point theorem, Critical growth, Ground state solution, Nehari manifold