  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-Amp\`ere type equations. In this
paper, we show the existence, regularity and uniqueness of solutions to this
Monge-Amp\`ere type equation in the case $p\geq 1$ by studying variational
properties for a family of Monge-Amp\`ere functionals. Moreover, the existence
and optimal global H\"older regularity in the case $p<1$ and $q\geq n$ is also
be discussed.
