Strictly convex solutions for singular Monge–Ampère equations with nonlinear gradient terms: existence and boundary asymptotic behavior

Abstract

A new existence criteria of strictly convex solutions is established for the singular Monge–Ampere equations $$\begin{aligned} \left\{ \begin{array}{l} det(D^2u)=b(x)f(-u)+g(|Du|)\ \text{ in } \Omega ,\\ u=0\ \text{ on } \partial \Omega , \end{array} \right. \end{aligned}$$ and $$\begin{aligned} \left\{ \begin{array}{l} det(D^2u)=b(x)f(-u)(1+g(|Du|))\ \text{ in } \Omega ,\\ u=0\ \text{ on } \partial \Omega . \end{array} \right. \end{aligned}$$ Under $$b,\ f$$ and g satisfying suitable conditions, we prove that the above boundary value problems admit a strictly convex solution, which turns out that this case is more difficult to handle than Monge–Ampere problems without gradient terms and needs some new ingredients in the arguments. Then we show the asymptotic behavior of strictly convex solutions under appropriate conditions. On the technical level, we adopt the sub-supersolution method and the Karamata regular variation theory.