Sharp conditions for the existence of boundary blow-up solutions to the Monge-Ampère equation (Q1751568)

For a particular class of Monge-Ampère equations, the authors provide criteria for existence/non-existence of strictly convex solutions which blow-up on the boundary. To state the main results only, let \(\Omega\) be a strictly convex, bounded, smooth domain of \(\mathbb{R}^n\), \(n\geq2\), and let \(K\in C^\infty(\Omega)\) be a positive function. Let also \(f\in C^\infty\big((\eta,\infty)\big)\), \(\eta\in[-\infty,\infty)\), be positive, strictly increasing, and such that \(f(\eta^+)=0\) in case \(\eta\neq-\infty\). Setting \[ F(s):= \begin{cases} \int_\eta^sf(t)dt & \text{if }\eta\neq-\infty \\ \int_0^sf(t)dt & \text{if }\eta=-\infty, \end{cases} \] assume further in case \(\eta\neq-\infty\), that \(\int_\eta^{\eta+\epsilon}\big(F(s)\big)^{-1/(n+1)}ds=\infty\) for all small \(\epsilon>0\). Sharpening several results in the literature, it is shown that the condition \[ \int_M^{\infty}\big(F(s)\big)^{-1/(n+1)}ds<\infty~\text{ for all large }M>0, \] {\parindent=0.7cm\begin{itemize}\item{\textbf{I.}} is equivalent to the existence of a strictly convex solution to the boundary value problem \[ (\star)~\begin{cases} \text{det}\big(D^2 u(x)\big)=K(x)f\big(u(x)\big) & \text{for }x\in\Omega \\ u(x)\rightarrow\infty & \text{as }\text{dist}(x,\partial\Omega)\rightarrow0, \end{cases} \] provided \(K\) is additionally bounded. \item{\textbf{II.}} implies the existence of a strictly convex solution to \((\star)\) provided that \(K\) is such that the Dirichlet boundary value problem \[ \begin{cases} \text{det}\big(D^2 u(x)\big)=K(x) & \text{for }x\in\Omega \\ u=0 & \text{on }\partial\Omega, \end{cases} \] has a strictly convex solution. \end{itemize}} Each of the above statements is not true in case \(\eta\neq-\infty\) and \(\int_\eta^{\eta+\epsilon}\big(F(s)\big)^{-1/(n+1)}ds<\infty\) for some \(\epsilon>0\).