A Singular Perturbation Problem for a Quasi-Linear Operator Satisfying the Natural Growth Condition of Lieberman

Abstract

In this paper we study the following problem. For ε > 0, take uε as a solution of Luε := div ( g(|∇uε|) |∇uε| ∇uε) = βε(uε), uε ≥ 0. A solution to (Pε) is a function uε ∈ W1,G(Ω)∩L∞(Ω) such that Ω g(|∇uε|) ∇uε |∇uε| ∇ϕ dx = − Ω ϕ βε(uε) dx for every ϕ ∈ C∞0 (Ω). Here βε(s) = 1 ε β s ε , with β ∈ Lip(R), β > 0 in (0, 1) and β = 0 otherwise. We are interested in the limiting problem, when ε → 0. As in previous work with L = Δ or L = Δp we prove, under appropriate assumptions, that any limiting function is a weak solution to a free boundary problem. Moreover, for nondegenerate limits we prove that the reduced free boundary is a C1,α surface. This result is new even for Δp. Throughout the paper, we assume that g satisfies the conditions introduced by Lieberman in [Comm. Partial Differential Equations, 16 (1991), pp. 311-361].