A weak solution to quasilinear elliptic problems with perturbed gradient

We consider weak solutions to the Dirichlet problem -divA(x,Du-Θ(u))=finΩ,u=0on∂Ω,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{ll} -\text {div}\,A\big (x,Du-\varTheta (u)\big )=f\quad &{}\text {in}\;\varOmega ,\\ u=0\quad &{}\text {on}\;\partial \varOmega , \end{array} \right. \end{aligned}$$\end{document}where Θ:Rm→Mm×n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varTheta :{\mathbb {R}}^m\rightarrow {\mathbb {M}}^{m\times n}$$\end{document} is a continuous function assumed to satisfy a Lipschitz condition. Based on the theory of Young measures, we prove the existence result when f∈W-1,p′(Ω;Rm)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\in W^{-1,p'}(\varOmega ;{\mathbb {R}}^m)$$\end{document}.