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

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 nonnegativecontinuous function on ∂D and f is a nonnegativefunction satisfying some appropriate conditions related to someKato class of functions K(D). Our aim is to prove that the aboveproblem has a continuous positive solution bounded below by afixed 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