On positive solutions of quasilinear elliptic systems

In this paper, we consider the existence and nonexistence of positive solutions of degenerate elliptic systems \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\left\{ \begin{gathered} - \Delta _p u = f(x,u,v), in \Omega , \hfill \\ - \Delta _p u = g(x,u,v), in \Omega , \hfill \\ u = v = 0, on \partial \Omega , \hfill \\ \end{gathered} \right.$$ \end{document} where −Δp is the p-Laplace operator, p > 1 and Ω is a C1,α-domain in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb{R}^n $$ \end{document}. We prove an analogue of [7, 16] for the eigenvalue problem with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$f(x,u,v) = {\lambda }_{1} v^{p - 1} ,{ }g(x,u,v) = {\lambda }_{2} u^{p - 1} $$ \end{document} and obtain a non-existence result of positive solutions for the general systems.