Multiple solutions for the Schrodinger equations with sign-changing potential and Hartree nonlinearity

In this paper, we study the following Schrodinger equation {-Delta u + V-lambda(x)u + mu phi vertical bar u vertical bar(p-2) u = f(x, u) + beta(x)vertical bar u vertical bar(nu-2)u, in R-3, (-Delta)(alpha/2) phi = mu vertical bar u vertical bar(p), in R-3, where mu >= 0 is a parameter, alpha is an element of (0, 3), nu is an element of (1, 2) and p is an element of [2, 3 + 2 alpha). V-lambda is allowed to be sign-changing and phi vertical bar u vertical bar(p-2) u is a Hartree-type nonlinearity. We require that V-lambda = lambda V+ - V- with V+ having a bounded potential well Omega whose depth is controlled by lambda. Under some mild conditions on V-lambda(x) and f (x, u), we prove that the above system has at least two nontrivial solutions. Specially, our results cover the general Schrodinger equations and the Schrodinger-Poisson equations. (C) 2017 Published by Elsevier Ltd.