Positive solutions for some non-autonomous Schrödinger-Poisson systems

In this paper we study the Schrödinger-Poisson system(SP)(- Δ u + u + K (x) φsymbol (x) u = a (x) | u |p - 1 u,, x ∈ R3,; - Δ φsymbol = K (x) u2,, x ∈ R3,) with p ∈ (3, 5). Assuming that a : R3 → R and K : R3 → R are nonnegative functions such thatunder(lim, | x | → ∞) a (x) = a∞ > 0, under(lim, | x | → ∞) K (x) = 0 and satisfying suitable assumptions, but not requiring any symmetry property on them, we prove the existence of positive solutions. © 2009 Elsevier Inc. All rights reserved