Blow-up solutions to the Monge–Ampère equation with a gradient term: sharp conditions for the existence and asymptotic estimates

We consider the boundary blow-up Monge–Ampère problem with a gradient term M[u]=K(x)f(u)|∇u|qforx∈Ω,u(x)→+∞asdist(x,∂Ω)→0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} M[u]=K(x)f(u)|\nabla u|^q \text{ for } x \in \Omega ,\; u(x)\rightarrow +\infty \text{ as } \mathrm{dist}(x,\partial \Omega )\rightarrow 0, \end{aligned}$$\end{document}where M[u]=det(uxixj)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M[u]=\det \, (u_{x_{i}x_{j}})$$\end{document} is the Monge–Ampère operator, q≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q\ge 0$$\end{document}, 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 smooth, bounded, strictly convex domain in RN(N≥2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathbb {R}^N \, (N\ge 2)$$\end{document}, and K, f are smooth positive functions. Under K and q satisfying suitable conditions, we first prove that the above boundary blow-up problem admits a strictly convex solution if and only if f satisfies a Keller–Osserman type condition. Then we show the asymptotic behavior of strictly convex solutions to the boundary blow-up Monge–Ampère problem under a new weaker condition than previous references. Finally, we also show the existence of strictly convex solutions under appropriate assumptions on K and q, without assuming that f satisfies a Keller–Osserman type condition. On the technical level, we adopt the sub-supersolution method and the Karamata regular variation theory.