Existence Results For A Class Of Nonlinear Degenerate Elliptic Equations

In this paper we are interested in the existence of solutions for Dirichlet problem associated with the degenerate nonlinear elliptic equations

{-div[𝒜(x,∇u)ω1+𝒝(x,u,∇u)ω2]=f0(x)-∑j=1nDjfj(x) in Ω,u(x)=0 on ∂Ω,\left\{ {\matrix{ { - {\rm{div}}\left[ {\mathcal{A}\left( {x,\nabla u} \right){\omega _1} + \mathcal{B}\left( {x,u,\nabla u} \right){\omega _2}} \right] = {f_0}\left( x \right) - \sum\limits_{j = 1}^n {{D_j}{f_j}\left( x \right)\,\,{\rm{in}}} \,\,\,\,\,\Omega ,} \hfill \cr {u\left( x \right) = 0\,\,\,\,{\rm{on}}\,\,\,\,\partial \Omega {\rm{,}}} \hfill \cr } } \right.

in the setting of the weighted Sobolev spaces.