Existence, uniqueness and multiplicity of positive solutions for Schrodinger-Poisson system with singularity

In this paper, we consider the following Schrodinger-Poisson system with singularity {-Delta u+eta phi u=mu u-r, in Omega, -Delta phi=u(2,) in Omega, u>0, in Omega, u=phi=0, on partial derivative Omega, where Omega C R-3 is a smooth bounded domain with boundary partial derivative Omega, eta = +/-1, r is an element of (0,1) is a constant, mu > 0 is a parameter. We obtain the existence and uniqueness of positive solution for n = 1 and any mu > 0 by using the variational method. The existence and multiplicity of solutions for the system are also considered for eta = -1 and mu > 0 small enough by using the method of Nehari manifold. (C) 2016 Elsevier Inc. All rights reserved.