Some remarks on a sign condition for perturbations of nonlinear problems

Abstract:

In this paper, we shall be concerned with the existence result of the quasilinear elliptic equations of the form,

, where A is a Leray-Lions operator from into its dual. On the nonlinear lower order term g(x, u, ∇u), we assume that it is a Carathéodory function having natural growth with respect to |∇u|, but without assuming the sign condition. The main novelty of our work is a new technique based on a Poincaré's inequality. The right hand side f belongs to W^-1,P′(Ω).