Positive solutions for some non-autonomous Schrödinger–Poisson systems

Abstract In this paper we study the Schrodinger–Poisson system (SP) { − Δ u + u + K ( x ) ϕ ( x ) u = a ( x ) | u | p − 1 u , x ∈ R 3 , − Δ ϕ = K ( x ) u 2 , x ∈ R 3 , with p ∈ ( 3 , 5 ) . Assuming that a : R 3 → R and K : R 3 → R are nonnegative functions such that lim | x | → ∞ a ( x ) = a ∞ > 0 , lim | x | → ∞ K ( x ) = 0 and satisfying suitable assumptions, but not requiring any symmetry property on them, we prove the existence of positive solutions.

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