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 $0\le -\Delta u\le (\frac{1}{|x{|}^{\alpha }}\ast v{)}^{\lambda }$ $0\le -\Delta v\le (\frac{1}{|x{|}^{\beta }}\ast u{)}^{\sigma }$ 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 $\alpha ,\beta \in (2,n+2)$.

32 pages, one figure. To be published in Calculus of Variations and PDE. Errors are corrected in the second paragraph of Step 2 on pages 22 and 23. Also corrections are made on page 27, line 9 from the bottom, and on page 29, line 3 from the bottom