Blow-up rates of large solutions for a ϕ-Laplacian problem with gradient term

Abstract

We study the behaviour of solutions of a boundary blow-up elliptic problem on a bounded domain Ω with smooth boundary in R N . The data of the problem consist of an increasing function f : R + → R + and two real regularly varying functions φ and g.

Asymptotic properties of a $\varphi$-Laplacian and Rayleigh quotient

“…Regularity (Hölder continuity), positivity and vanishing at infinity of the solutions have subsequently been proved in [4]. In the article [2] the asymptotic properties of blow-up (large) solutions of (1.1) have been treated, when the right-hand side satisfies particular growth conditions and the left-hand side contains an additional nonlinear term in |∇u|. In all these references we assume a Lieberman-like condition, see [22, (1.1)], but the hypothesis of differentiability on ϕ is dropped.…”

Regularity, positivity and asymptotic vanishing of solutions of a φ-Laplacian

“…where φ : R → R is an odd, increasing and not-necessarily differentiable homeomorphism, see [4,9,10]. For example, in [3] we have treated the blow-up rates of large solutions of such φ-Laplacians via a dynamical formulation. The latter provides a parallel analysis between the asymptotic properties of large solutions and the behavior of the orbits of an autonomous ordinary differential equation.…”