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

Let D be a bounded domain in ℝ n ( n ≥ 2 ) . We consider the following nonlinear elliptic problem: Δ u = 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 Δ u = − 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.