Boundary blow-up in nonlinear elliptic equations of Bieberbach–Rademacher type

Abstract

Abstract. We establish the uniqueness of the positive solution for equations of the form −∆u = au − b(x)f (u) in Ω, u| ∂Ω = ∞. The special feature is to consider nonlinearities f whose variation at infinity is not regular (e.g., exp(u) − 1, sinh(u), cosh(u) − 1, exp(u) log(u + 1), u β exp(u γ ), β ∈ R, γ > 0 or exp(exp(u)) − e) and functions b ≥ 0 in Ω vanishing on ∂Ω. The main innovation consists of using Karamata's theory not only in the statement/proof of the main result but also to link the nonregular variation of f at infinity with the blow-up rate of the solution near ∂Ω.

Asymptotic Behavior and Uniqueness of Blow-up Solutions of Elliptic Equations

“…For example, [16] is concerned with the blow-up problem for the equation div(|Du| p−2 Du) = g(u), and [8,9,7] look at the equation ∆u + au = b(x)g(u) with a a suitable constant and b a nonnegative function satisfying some additional technical conditions relating b to g and a. We defer a study of such problems to a future work, but point out here that we are able to study g from a larger class of functions than in those works (when specialized to p = 2 in [16] and to b ≡ 1 and a = 0 in [8,9,7]). …”

“…This example was inspired by the structure conditions in [9]. There are several important differences to note, however.…”

“…There are several important differences to note, however. In [9], the equation has the form ∆u+au = b(x)g(u), and the function g has the form exp m (h) with exp m being the m-th iterated exponential (exp 1 = exp and exp m = exp(exp m−1 )) and h being a function of regular variation. Unlike the situation for our previous example, this class of gs is not contained in ours even if we make the straightforward modifications needed to replace the exponential by an iterated exponential.…”

“…Unlike the situation for our previous example, this class of gs is not contained in ours even if we make the straightforward modifications needed to replace the exponential by an iterated exponential. On the other hand, we allow functions that cannot be written in the form used in [9].…”

Existence and asymptotic behavior of positive solutions of a semilinear elliptic system in a bounded domain

“…We study in this paper system (1.1) in a more general situation that treat the cases a 1 , a 2 = 1 with no restriction on the sign of the exponents. Our approach relies on the asymptotic behavior of solutions to the following singular elliptic problem 5) where α < 1 and a satisfies…”