Regularity of the Extremal Solutions in a Gelfand System Problem

Abstract

Abstract We examine the elliptic system given by where λ, γ are positive parameters and where Ω is a smooth bounded domain in ℝN. Let U denote the parameter region (λ, γ) of strictly positive parameters where (P)λ,γ has a smooth solution and let Υ denote the boundary of U. We show that the extremal solution (u∗, v∗) associated with (λ∗, γ∗) ∈ Υ is smooth provided that 3 ≤ N ≤ 9 and