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 C² positive solutions u(x) and v (x) of the system

$$\matrix{{0 \le - {\rm{\Delta}}u \le f(v)} \\ {0 \le - {\rm{\Delta}}v \le g(u)} \\} \quad {\rm{in}}\,{B_1}\left(0 \right)\,\backslash \left\{0\right\}\, \subset {\mathbb{R}^n},\,n \ge 2,$$

where f, g:(0, ∞) → (0, ∞) are continuous functions. We provide optimal conditions on f and g at ∞ 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 ≥ 3 and under the assumption f (t) = O(t^λ), g(t) = O(t^σ)as t → ∞ (λ, σ ≥ 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.