Regularities for solutions to the $L_p$ dual Minkowski problem for unbounded closed sets

Recently, the $L_p$ dual Minkowski problem for unbounded closed convex sets in a pointed closed convex cone was proposed and a weak solution to this problem was provided. In smooth setting, this problem is equivalent to solving the Dirichlet problem for a class of Monge-Ampère type equations. In this paper, we show the existence, regularity and uniqueness of solutions to this Monge-Ampère type equation in the case $p\ge 1$ by studying variational properties for a family of Monge-Ampère functionals. Moreover, the existence and optimal global Hölder regularity in the case $p<1$ and $q\ge n$ is also be discussed.