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

We study the behavior near the origin of $C^2$ positive solutions $u(x)$ and $v(x)$ of the system $0\le -\mathrm{\Delta }u\le (\frac{1}{\mathrm{\mid }x{\mathrm{\mid }}^{\alpha }}\ast v{)}^{\lambda }$ $0\le -\mathrm{\Delta }v\le (\frac{1}{\mathrm{\mid }x{\mathrm{\mid }}^{\beta }}\ast u{)}^{\sigma }$ 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$\alpha, \beta\in (2,n+2)$.