Infinitely many solutions for Schrodinger-Newton equations

We prove the existence of infinitely many non-radial positive solutions for the Schro spexpressioncing diexpressioneresis dinger-Newton system { delta u - V (|x|)u + Psi u =0, x is an element of R-3, delta Psi + 1/2 u(2) = 0, x is an element of R-3, provided that V (r) has the following behavior at infinity: a V (r) = V-0 +a/r(m) +O (1 /r (m)+theta as r -> infinity, where 1/2 <= m < 1 and a, V-0,V-theta are some positive constants. In particular, for any s large we use a reduction method to construct s-bump solutions lying on a circle of radius r similar to(s logs)( 1) 1-m .