Nonlinear Elliptic Boundary Value Problems by Topological Degree

Abstract

Based on the recent Berkovits degree, and by way of an abstract Hammerstein equation, we study the Dirichlet boundary value problem involving nonlinear operators of the form $$ - div a(x,\nabla u)=\lambda u+f(x, u,\nabla u),$$ where a and f are Caratheodory functions satisfying some nonstandard growth conditions and \(\lambda \in I\!R\) . The function a satisfy also a condition of strict monotony and a condition of coercivity. We prove the existence of weak solutions of this problem in the weighted Sobolev spaces \(W_0^{1,p(x)}(\varOmega ,\rho )\) where \(\rho \) is a weight function, satisfying some integrability conditions.