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Abstract and Applied Analysis
Volume 2006, Article ID 95480, 18 pages
http://dx.doi.org/10.1155/AAA/2006/95480

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

Faten Toumi

Département de Mathématiques, Faculté des Sciences de Tunis, Campus Universitaire, Tunis 2092, Tunisia

Received 10 June 2004; Accepted 22 September 2004

Copyright © 2006 Faten Toumi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.