Existence of solutions for unilateral problems associated to some quasilinear anisotropic elliptic equations with measure data

Abstract

In this paper, we will study the existence of solutions for some nonlinear anisotropic elliptic equation of the type {Au+g(x,u,∇u)=μ−div φ(u)in Ω,u=0on ∂Ω,\left\{ {\matrix{{Au + g\left( {x,u,\nabla u} \right) = \mu - div\,\phi \left( u \right)} \hfill & {in\,\Omega ,} \hfill \cr {u = 0} \hfill & {on\,\,\partial \Omega ,} \hfill \cr } } \right. where Au=−∑i=1N∂∂xiai(x,u,∇u)Au = - \sum\limits_{i = 1}^N {{\partial \over {\partial {x_i}}}{a_i}\left( {x,u,\nabla u} \right)} is a Leray-Lions operator, the Carathéodory function g(x, s, ξ) is a nonlinear lower order term that verify some natural growth and sign conditions, where the data µ = f − div F belongs to L1−dual and ϕ (·) ∈ C0(R, RN).

Keywords

• unilateral problem
• nonlinear elliptic equations
• anisotropic sobolev spaces
• entropy solutions
• measure data
• 35j15
• 35j62