  General $L_p$ dual curvature measures have recently been introduced by
Lutwak, Yang and Zhang. These new measures unify several other geometric
measures of the Brunn-Minkowski theory and the dual Brunn-Minkowski theory.
$L_p$ dual curvature measures arise from $q$th dual inrinsic volumes by means
of Alexandrov-type variational formulas. Lutwak, Yang and Zhang formulated the
$L_p$ dual Minkowski problem, which concerns the characterization of $L_p$ dual
curvature measures. In this paper, we solve the existence part of the $L_{p}$
dual Minkowski problem for $p>1$ and $q>0$, and we also discuss the regularity
of the solution.
