Ground state solutions for the nonlinear Schrödinger-Maxwell equations

In this paper we study the nonlinear Schrödinger-Maxwell equations{(- Δ u + V (x) u + φ{symbol} u = | u |p - 1 u, in R3,; - Δ φ{symbol} = u2, in R3 .) If V is a positive constant, we prove the existence of a ground state solution (u, φ{symbol}) for 2 < p < 5. The non-constant potential case is treated for 3 < p < 5, and V possibly unbounded below. Existence and nonexistence results are proved also when the nonlinearity exhibits a critical growth.