On the existence of bounded solutions of nonlinear elliptic systems

We study the existence of bounded solutions to the elliptic system−Δpu=f(u,v)+h1inΩ,−Δqv=g(u,v)+h2inΩ,u=v=0on∂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 operatorTand some invariant setKwhere we can use the Leray Schauder's theorem.

Existence of a solution of a Fourier nonlocal quasilinear parabolic problem

The aim of this paper is to give a theorem about the existence of a classical solution of a Fourier third nonlocal quasilinear parabolic problem. To prove this theorem, Schauder's theorem is used. The paper is a continuation of papers [1]-[8] and the generalizations of some results from [9]-[11]. The theorem established in this paper can be applied to describe some phenomena in the theories of diffusion and heat conduction with better effects than the analogous classical theorem about the existence of a solution of the Fourier third quasilinear parabolic problem.