Positive solutions for some non-autonomous Schrödinger–Poisson systems

By G. Cerami, Giusi Vaira. 2010

406 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/395671a7de16c39fdba3afb13adb2d5da5a64839

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Citations in the corpus

On ground state solutions for the Schrödinger–Poisson equations with critical growth (2014, neutral): “[11] G.”
On ground state solutions for some non-autonomous Schrödinger–Poisson systems☆ (2012, background): “Very recently, Cerami and Vaira [10] studied system (SP) with f (u) = |u|p−1u and 3 < p < 5.”
Fractional Schrödinger–Poisson Systems with a General Subcritical or Critical Nonlinearity (2015, positive): “6 [10] G. Cerami, G. Vaira, Positive solutions for some non-autonomous Schrödinger-Poisson systems, J. Differential Equations 248 (2010), 521–543.”
Existence of least energy nodal solution for a Schrödinger–Poisson system in bounded domains (2013, background): “[13] G. Cerami & G. Vaira, Positive solutions for some non-autonomous Schrödinger-Poisson systems, J. Differential Equations 248 (2010) 521– 543.”
Existence and concentration of positive solutions for semilinear Schrödinger–Poisson systems in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin} (2012, neutral): “Proof As in [14,35], one can easily check the conclusions (i) and (i i).”