Asymptotic behavior at isolated singularities for solutions of nonlocal semilinear elliptic systems of inequalities

TL;DR

This paper investigates the asymptotic behavior of positive solutions to nonlocal semilinear elliptic systems near isolated singularities, providing new insights into their local structure and related integral systems.

Contribution

It introduces novel methods to analyze the asymptotic behavior of solutions to nonlocal elliptic systems and their integral counterparts near singularities.

Findings

Characterization of solution behavior near the origin

Results on the asymptotics of solutions to related integral systems

Development of techniques applicable to nonlocal elliptic inequalities

Abstract

We study the behavior near the origin of $C^2$ positive solutions $u(x)$ and $v(x)$ of the system $0\le -\Delta u \le (\frac{1}{|x|^\alpha}* v)^\lambda$ $0\le -\Delta v \le (\frac{1}{|x|^\beta}* u)^\sigma$ in $B_2(0)\setminus\{0\} \subset R^n$, $n\ge 3$, where $\lambda,\sigma \ge 0$ and $\alpha,\beta\in (0,n)$. A by-product of our methods used to study these solutions will be results on the behavior near the origin of $L^1(B_1(0))$ solutions $f$ and $g$ of the system $0 \le f(x) \le C(|x|^{2-\alpha} + \int_{|y|<1}\frac{ g(y) dy}{|x-y|^{\alpha-2}} )^\lambda$ $0 \le g(x) \le C(|x|^{2-\beta} + \int_{|y|<1}\frac{ f(y) dy}{|x-y|^{\beta-2}} )^\sigma$ for $0<|x|<1$ where $\lambda,\sigma \ge 0$ and $\alpha, \beta\in (2,n+2)$.