Estimates on the Green Function and Existence of Positive Solutions of Nonlinear Singular Elliptic Equations

Abstract

We establish a 3G-Theorem for the Green's function for an unbounded regular domain D in ℝn(n ≥ 3), with compact boundary. We exploit this result to introduce a new class of potentials K(D) that properly contains the classical Kato class [Formula: see text]. Next, we study the existence and the uniqueness of a positive continuous solution u in [Formula: see text] of the following nonlinear singular elliptic problem [Formula: see text] where φ is a nonnegative Borel measurable function in D × (0, ∞), that belongs to a convex cone which contains, in particular, all functions φ(x, t) = q(x)t-σ, σ ≥ 0 with q ∈ K(D). We give also some estimates on the solution u.