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

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), uᵝ exp(uᵞ), 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.