Multiple solutions for a nonhomogeneous Schrodinger-Maxwell system in R3

The paper considers the following nonhomogeneous Schrodinger-Maxwell system: {-Delta u + u + lambda phi(x)u = vertical bar u vertical bar(p-1)u + g(x), x is an element of R-3, -Delta phi = u(2), x is an element of R-3, (SM) where lambda > 0, p is an element of (1, 5), and 0 <= g(x) = g(vertical bar x vertical bar) is an element of L-2(R-3). There seem to be no results on the existence of multiple solutions to problem (SM) for p is an element of (1, 3). In this paper, we find that there is a constant C-p > 0 such that problem (SM) has at least two solutions for all p is an element of (1, 5) provided that parallel to g parallel to(L2) <= C-p, however, for p is an element of (1, 2] we need lambda > 0 is small. Moreover, C-p = (p-1)/2p [(p+1)Sp+1/2p](1/(p-1)), where S is the Sobolev constant. (C) 2013 Elsevier Ltd. All rights reserved.