Pointwise Bounds and Blow-up for Systems of Semilinear Elliptic Inequalities at an Isolated Singularity via Nonlinear Potential Estimates

Abstract

We study the behavior near the origin of $C^2$ positive solutions $u(x)$ and $v(x)$ of the system $0\le -\Delta u\le f(v)$ $0\le -\Delta v\le g(u)$ in $B_1(0)\backslash \{0\}$ where $f,g:(0,\infty )\to (0,\infty )$ are continuous functions. We provide optimal conditions on $f$ and $g$ at $\infty$ such that solutions of this system satisfy pointwise bounds near the origin. In dimension $n=2$ we show that this property holds if ${\log }^{+}f$ or ${\log }^{+}g$ grow at most linearly at infinity. In dimension $n\ge 3$ and under the assumption $f(t)=O(t^{\lambda })$, $g(t)=O(t^{\sigma })$ as $t\to \infty$, ($\lambda ,\sigma \ge 0$), we obtain a new critical curve that optimally describes the existence of such pointwise bounds. Our approach relies in part on sharp estimates of nonlinear potentials which appear naturally in this context.

arXiv comment41 pages, 1 figure