Uniqueness and asymptotic behaviour for solutions of semilinear problems with boundary blow-up

Abstract

Abstract. In this paper we prove uniqueness of positive solutions to logistic singular problemswhere the main feature is the fact that a |∂Ω = 0. More importantly, we provide exact asymptotic estimates describing, in the form of a two-term expansion, the blow-up rate for the solutions near ∂Ω. This expansion involves both the distance function d(x) = dist(x, ∂Ω) and the mean curvature H of ∂Ω.

Asymptotic behaviour and symmetry of positive solutions to nonlinear elliptic equations in a half-space

“…We shall also approximately describe the behaviour of positive solutions of (1.7) near the origin and at infinity. Here and in the following sections, we need to use some arguments of elliptic equations with boundary blowup conditions, which can be found in [14,17,21,22]. Proposition 2.3.…”

Multiplicity of positive solutions to boundary blow‐up problem with variable exponent and sign‐changing weights

“…This type of problem has been extensively studied in the past two decades. We refer to for the case that a ( x ) is bounded below from zero, and p ( x ) is a constant, and for the case that a ( x ) positive in Ω but can vanish or be singular on ∂ Ω. Weights that can also vanish in Ω have been considered in .…”

The solvability of an elliptic system under a singular boundary condition

“…We quote the pioneering papers [5], [33], [32] and [29] concerning Riemannian geometry and Riemann surfaces and [23] and [24], where (1.1) arises in a problem in electrohydrodynamics (see also [26], where stochastic control problems lead to large solutions). A brief account of more recent literature on the problem is provided by [25], [2], [36], [3], [10], [27], [28], [34], [1], [31], [4], [37], [9] and [19].…”

“…The extension to equations involving the p-Laplacian was first considered in [10]. In [19] the case where a vanishes on ∂Ω was studied, while a is even allowed to be unbounded near ∂Ω in [37], [6], [7] and [14] (see an updated account in [15] and the references included there on this issue). It is also worth remarking that [18] gives a study case where problem (1.1) arises in population dynamics.…”

“…It is also worth remarking that [18] gives a study case where problem (1.1) arises in population dynamics. Specifically, −∆u = λ(x)u − a(x)u p in Ω, u = +∞ on ∂Ω with p > 1, λ, a Hölder continuous in Ω, a > 0 in Ω but a |∂Ω = 0 (see also [19]).…”