A priori estimates and existence for quasi-linear elliptic equations

We study the boundary value problem of quasi-linear elliptic equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{array}{rl} {\rm div}(|\nabla u|^{m-2} \nabla u) + B(z,u,\nabla u) = 0 &\quad {\rm in}\, \Omega,\\ u = 0 &\quad {\rm on} \,\partial\Omega, \end{array}$$\end{document}where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Omega\subset\mathbb{R}^n}$$\end{document} (n ≥ 2) is a connected smooth domain, and the exponent \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${m\in(1,n)}$$\end{document} is a positive number. Under appropriate conditions on the function B, a variety of results on a priori estimates, existence and non-existence of positive solutions have been established. The results are generically optimum for the canonical prototype B = |u|p-1u, p > m − 1.