Isolated singularities of positive solutions for Choquard equations in sublinear case

2016/10/28 by Huyuan Chen, Feng Zhou, Chen, Huyuan +1 · 1 citation
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP

paper · pdf · doi:10.48550/arxiv.1610.09062

16 pages

arxiv created 2017/03/07 · arxiv updated 2017/03/08

Abstract

Our purpose of this paper is to study the isolated singularities of positive solutions to Choquard equation in the sublinear case q ∈ (0,1) -Δu+ u =I_α[u^p] u^q \rm in ℝ^N∖\0\, % [2mm] \phantom lim_|x|→+∞u(x)=0, where p >0, N ≥ 3, α∈ (0,N) and I_α[u^p](x) = ∫_ℝ^N \fracu^p(y)|x-y|^N-αdy is the Riesz potential, which appears as a nonlocal term in the equation. We investigate the nonexistence and existence of isolated singular solutions of Choquard equation under different range of the pair of exponent (p,q). Furthermore, we obtain qualitative properties for the minimal singular solutions of the equation.

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