Asymptotic behavior at isolated singularities for solutions of nonlocal semilinear elliptic systems of 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 $\alpha, \beta\in (2,n+2)$.