Extremal solutions for nonvariational quasilinear elliptic systems via expanding trapping regions

We consider the Dirichlet boundary value problem for quasilinear elliptic systems in a bounded domain Ω⊂RN\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} with a diagonal (p1,p2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(p_1, p_2)$$\end{document}-Laplacian as leading differential operator of the form -Δpiui=fi(x,u1,u2,∇u1,∇u2)inΩ,ui=0on∂Ω,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} -\Delta _{p_i} u_i=f_i(x, u_1,u_2,\nabla u_1,\nabla u_2)\ \ \text {in }\Omega ,\ \ u_i=0\ \ \text {on }\partial \Omega , \end{aligned}$$\end{document}where the component functions fi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_i$$\end{document} (i=1,2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i=1,2$$\end{document}) of the lower order vector field may also depend on the gradient of the solution u=(u1,u2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u=(u_1,u_2)$$\end{document}. The main goal of this paper is twofold. First, we establish an enclosure and existence result by means of the trapping region which is formed by pairs of appropriately defined sub-supersolutions. Second, by a suitable construction of sequences of expanding trapping regions we are able to prove the existence of extremal positive and negative solutions of the system. The theory of pseudomonotone operators, regularity results due to Cianchi-Maz’ya, as well as a strong maximum principle due to Pucci-Serrin are essential tools in the proofs.