Pointwise bounds and blow-up for systems of semilinear elliptic inequalities at an isolated singularity via nonlinear potential estimates

摘要

We study the behavior near the origin of C2 positive solutions u(x) and v (x) of the system 0≤?Δu≤f(v)0≤?Δv≤g(u)inB1(0)(0)??n,n≥2,documentclass[12pt]{minimal}usepackage{amsmath}usepackage{wasysym}usepackage{amsfonts}usepackage{amssymb}usepackage{amsbsy}usepackage{mathrsfs}usepackage{upgreek}setlength{oddsidemargin}{-69pt}egin{document}$$matrix{{0 le - {m{Delta}}u le f(v)} {0 le - {m{Delta}}v le g(u)} } quad {m{in}},{B_1}left(0 ight),ackslash left{0ight}, subset {mathbb{R}^n},,n ge 2,$$end{document} 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(tλ), g(t) = O(tσ)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.