Schrödinger-Poisson System Involving Potential Vanishing at Infinity and Unbounded Below

This article concerns the following class of system { -triangle u + V(x)u + & ell;(x)phi u = f(u) + lambda|u|(q-2 )u in R-3, -triangle phi = & ell;(x)u(2) in R-3, u, phi is an element of D-1,D-2(R-3), u,phi >= 0 in R-3, where lambda >= 0 and q >= 2(& lowast;) = 6 is the critical Sobolev exponent in dimension 3, the nonlinearity f : R -> R is superlinear and has sub critical growth, V, & ell; : R-3 -> R are measurable functions with & ell; is an element of L-2(R-3), the potential V can change sign in R-3 and vanish at infinity, that is, V (x)-> 0 as |x|->infinity. Our approach is based on variational method combined with Benci-Fortunato's reduction argument [Topol. Methods Nonlinear Anal. 11 (1998) 283-293], Del Pino-Felmer's penalization technique [Calc. Var. Partial Diff. Equations 4 (1996) 121-137] and L-infinity-estimate.