On the existence of bounded solutions of nonlinear elliptic systems

Abstract

We study the existence of bounded solutions to the elliptic system −Δpu=f(u,v)+h1 in Ω, −Δqv=g(u,v)+h2 in Ω, u=v=0 on ∂Ω, non-necessarily potential systems. The method used is a shooting technique. We are concerned with the existence of a negative subsolution and a nonnegative supersolution in the sense of Hernandez; then we construct some compact operator T and some invariant set K where we can use the Leray Schauder's theorem.