A note on Schrodinger-Newton systems with decaying electric potential

We prove the existence of solutions for the singularly perturbed Schrodinger-Newton system {h(2) Delta psi - V(x)psi + U psi = 0 h(2) Delta U + 4 pi gamma vertical bar psi vertical bar(2) = 0 in R-3 with an electric potential V that decays polynomially fast at infinity. The solution psi concentrates, as h -> 0, around (structurally stable) critical points of the electric potential. As a particular case, isolated strict extrema of V are allowed. (C) 2010 Elsevier Ltd. All rights reserved.