Isolated singularities of positive solutions for Choquard equations in sublinear case

TL;DR

This paper investigates the existence, nonexistence, and qualitative properties of isolated singular solutions to a nonlocal Choquard equation with sublinear nonlinearity in the entire space, revealing conditions under which solutions exist or do not.

Contribution

It provides new criteria for the existence and nonexistence of isolated singular solutions to the Choquard equation in the sublinear case and analyzes their qualitative properties.

Findings

Identified parameter ranges for existence of solutions.

Proved nonexistence of solutions in certain regimes.

Characterized minimal singular solutions qualitatively.

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.