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

2014/02/01 by Marius Ghergu, Ghergu, Marius, Steven D. Taliaferro +3
Mathematics · #35B40 #35J47 #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP #msc:35B40 #msc:35J47

paper · pdf · doi:10.48550/arxiv.1402.0113

41 pages, 1 figure

arxiv created 2014/02/01 · arxiv updated 2014/02/04

Abstract

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) in B₁(0)\backslash\0\ 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.

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