Existence of ground state solutions for asymptotically linear Schrödinger–Poisson systems. (10th November 2015) - Du, Miao; Zhang, Fubao; Tian, Lixin
Abstract : In this paper, we study the following Schrödinger–Poisson system: − Δ u + V ( x ) u + λK ( x ) ϕu = f ( x, u ), x ∈ R 3, − Δ ϕ = K ( x ) u 2, x ∈ R 3, (SP) where λ > 0 is a parameter, K ∈ L p ( R 3 ) with 2≤ p ≤+ ∞, and the function f ( x, s ) may not be superlinear in s near zero and is asymptotically linear with respect to s at infinity. Under certain assumptions on V, K, and f, we give the existence and nonexistence results via variational methods. More precisely, when p ∈[2, + ∞ ), we obtain that system (SP) has a positive ground state solution for λ small; when p =+ ∞, we prove that system (SP) has a positive solution for λ small and has no any nontrivial solution for λ large. Copyright © 2015 John Wiley & Sons, Ltd.