Uniqueness and asymptotic behavior of solutions with boundary blow-up for a class of nonlinear elliptic equations

Abstract

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Uniqueness for boundary blow-up problems with continuous weights

“…Its study seems to have been started in [21], where the case a(x) = 1, p = (N + 2)/ (N − 2) was analyzed, and later continued in [19], [1], [2], [25], [24] (see also [10] for the p-Laplacian analogue), where a(x) ≥ a 0 > 0 in Ω and a linear term −λu is added in the equation in some cases. In all of them, uniqueness was obtained by means of precise boundary estimates, which took the form u ∼ Ad −α as d → 0, where A is given in terms of p and α = 2/(p − 1).…”

On Uniqueness of Large Solutions of Nonlinear Parabolic Equations in Nonsmooth Domains

“…We conjecture that the necessary and sufficient conditions, obtained by Dhersin-Le Gall when q = 2 [4] and Labutin [6] in the general case q > 1, and expressed by mean of a Wiener type criterion involving the C R N 2,q ′ -Bessel capacity, are still valid. As in [7], it is clear that if ∂Ω satisfies the exterior segment property and 1 < q < (N − 1)/(N − 3), then u 0 is a large solution.…”

“…Marcus and Véron prove in [7] that, there exists at most one large solution, provided ∂Ω is locally the graph of a continuous function. The aim of this article is to extend these questions to the parabolic equation u(x, t) = ∞ ∀(y, s) ∈ Γ × (0, ∞).…”

Large solutions for a class of semilinear integro-differential equations with censored jumps

“…After the works by Keller and Osserman, a broad variety of very interesting results concerning existence, uniqueness and asymptotic behavior near the boundary for large solutions of second-order reaction-diffusion equations have been obtained in the PDE framework using different techniques, see [3,12,14,25,26,27,29] for a nonexhaustive list of references. It is also worth to mention the deep connection of problems like (1.3) with stochastic superprocesses and the so-called Brownian snake, see [15,24] and references therein.…”