Multiple positive solutions of a class of non autonomous Schrodinger-Poisson systems

For the nonlinear Schrodinger equation coupled with Poisson equation of the version -Delta u+u+phi u=a(x) vertical bar u vertical bar(p-2) u+lambda k(x)u in R-3 and -Delta phi = u(2) in R-3, we prove the existence of two positive solutions in H-1 (R-3) when a(x) is sign changing and the linear part is not coercive. We show that the coupled term phi u is helpful to find multiple positive solutions when a(x) is sign changing, which gives striking contrast to the known result where phi u is proven to be an obstacle to get the existence of nontrivial solutions. Surprisingly we show that the term phi u can play the role similar to a sign condition f a(x)e(1)(p)dx < 0, which has turned out to be a necessary condition to the existence of positive solutions for semilinear elliptic equations with indefinite nonlinearity (see e.g. Alama et al. (1993)). (C) 2014 Elsevier Ltd. All rights reserved.