Abstract and Applied Analysis
Volume 2006 (2006), Article ID 95480, 18 pages
doi:10.1155/AAA/2006/95480

Existence of positive solutions for nonlinear boundary value problems in bounded domains of ℝn

Abstract

Let D be a bounded domain in ℝn(n≥2). We consider the following nonlinear elliptic problem: Δu=f(⋅,u) in D (in the sense of distributions), u|∂D=ϕ, where ϕ is a nonnegative continuous function on ∂D and f is a nonnegative function satisfying some appropriate conditions related to some Kato class of functions K(D). Our aim is to prove that the above problem has a continuous positive solution bounded below by a fixed harmonic function, which is continuous on D¯. Next, we will be interested in the Dirichlet problem Δu=−ρ(⋅,u) in D (in the sense of distributions), u|∂D=0, where ρ is a nonnegative function satisfying some assumptions detailed below. Our approach is based on the Schauder fixed-point theorem.