Nonlinear Elliptic Systems with Coupled Gradient Terms

In this paper, we analyze the existence and non-existence of nonnegative solutions to a class of nonlinear elliptic systems of type: {−Δu=|∇v|q+λfin Ω,−Δv=|∇u|p+μgin Ω,u=v=0on ∂Ω,u,v≥0in Ω,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \left \{ \textstyle\begin{array}{r@{\quad }c@{\quad }l@{\quad }l@{\quad }l} -\Delta u & = & |\nabla v|^{q}+\lambda f & \text{in } &\Omega , \\ -\Delta v& = &|\nabla u|^{p}+\mu g &\text{in } &\Omega , \\ u=v&=& 0 & \text{on } &\partial \Omega , \\ u,v& \geq & 0 & \text{in } &\Omega , \end{array}\displaystyle \right . $$\end{document} where Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Omega $\end{document} is a bounded domain of RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{R}^{N}$\end{document} and p,q≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p, q\ge 1$\end{document}. f,g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f,g$\end{document} are nonnegative measurable functions with additional hypotheses and λ,μ≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\lambda , \mu \ge 0$\end{document}.