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arXiv · 1610.09062

Isolated singularities of positive solutions for Choquard equations in sublinear case

Huyuan Chen·Feng Zhou
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Abstract

Our purpose of this paper is to study the isolated singularities of positive solutions to Choquard equation in the sublinear case $q \in (0,1)$ $$\displaystyle \ \ -\Delta u+ u =I_\alpha[u^p] u^q\;\; {\rm in}\; \mathbb{R}^N\setminus\{0\}, % [2mm] \phantom{ } \;\; \displaystyle \lim_{|x|\to+\infty}u(x)=0, $$ where $p >0, N \geq 3, \alpha \in (0,N)$ and $I_{\alpha}[u^p](x) = \int_{\mathbb{R}^N} \frac{u^p(y)}{|x-y|^{N-\alpha}}dy$ 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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Huyuan Chen, Feng Zhou. 2016-10-28. Isolated singularities of positive solutions for Choquard equations in sublinear case. https://arxiv.org/abs/1610.09062

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