Nonexistence results for the schrödinger-poisson equations with spherical and cylindrical potentials in double-struck R sign 3

Abstract

We study the following Schrödinger-Poisson system: - Δ u + V (x) u + φ u = | u | p - 1 u, - Δ φ = u 2, lim | x | → + ∞ φ (x) = 0, where u, φ: double-struck R sign 3 → double-struck R sign are positive radial functions, p ∈ (1, + ∞), x = (x 1, x 2, x 3) ∈ double-struck R sign 3, and V (x) is allowed to take two different forms including V (x) = 1 / (x 1 2 + x 2 2 + x 3 2) α / 2 and V (x) = 1 / (x 1 2 + x 2 2) α / 2 with α > 0. Two theorems for nonexistence of nontrivial solutions are established, giving two regions on the α - p plane where the system has no nontrivial solutions. © 2013 Yongsheng Jiang et al.