Existence of solutions to a strongly nonlinear elliptic coupled system of finite order

The existence of a capacity solution to the strongly nonlinear degenerate problem, namely, H(theta)+g(x,theta)=sigma(theta)|del psi|2,div(sigma(theta)del psi)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H(\theta )+g(x,\theta )=\sigma (\theta )|\nabla \psi |<^>{2}, {\text {div}}(\sigma (\theta ) \nabla \psi )=0$$\end{document} in Omega\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document} where g(x,theta)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g(x,\theta )$$\end{document} is a lower order term satisfies the sign condition but without any restriction on its growth and the operator H is of the form H(theta)=& sum;|nu|=0r(-1)|nu|D nu h nu x,D gamma theta,|gamma|<=|nu|,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} H (\theta )=\sum _{|\nu |=0}<^>{r}(-1)<^>{|\nu |} D<^>\nu \left( h_\nu \left( x, D<^>\gamma \theta \right) \right) , \quad |\gamma | \le |\nu |, \end{aligned}$$\end{document}is proved in the framework of Sobolev space of finite order.