Ground state solutions for the nonlinear Schrödinger–Maxwell equations

In this paper we study the nonlinear Schrödinger–Maxwell equations − u + V (x)u + φu = |u|p−1u in R3, − φ = u2 in R3. If V is a positive constant, we prove the existence of a ground state solution (u,φ) 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.