Existence and boundary asymptotic behavior of strictly convex solutions for singular Monge–Ampère problems with gradient terms

Abstract

In this paper, we study the existence as well as the boundary asymptotic behavior of strictly convex solutions for singular Monge–Ampère problems with gradient terms. The standard tools are Karamata regular variation theory and the sub-super-solution method. In order to apply these methods, we need to know the properties of the weight function $b$ and the nonlinear term $f$. We find new structure conditions on $b$ and $f$ to overcome the difficulties due to the singularity of $b$ and the gradient terms.