Existence of Solutions for Degenerate Elliptic Problems in Weighted Sobolev Space

Abstract

This paper is devoted to the study of the existence of solutions to a general elliptic problemAu+g(x,u,∇u)=f-div⁡F, withf∈L1(Ω)andF∈∏i=1NLp'(Ω,ωi*), whereAis a Leray-Lions operator from a weighted Sobolev space into its dual andg(x,s,ξ)is a nonlinear term satisfyinggx,s,ξsgn⁡(s)≥ρ∑i=1Nωi|ξi|p,|s|≥h>0, and a growth condition with respect toξ. Here,ωi,ωi*are weight functions that will be defined in the Preliminaries.

Solutions to p(x)-Laplace type equations via nonvariational techniques

“…We would like to notice that there are many papers dealt with equation of the form (1.1) in which nonlinearity is given under the natural growth of order p via monotone operator methods and approximation methods (see, e.g., [2,11,13,23] and references therein). However, this is not the case for the equations of the form (1.1) involving variable exponent of nonlinearity, that is, growth of order p(x).…”