Toumi, Faten
January 2006
Abstract & Applied Analysis;2006, p1
Academic Journal
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 ? is a nonnegative function satisfying some assumptions detailed below. Our approach is based on the Schauder fixed-point theorem.


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