In the paper ^[20], the authors introduced a Gauss curvature flow to study the Aleksandrov problem and the dual Minkowski problem. The paper ^[20] treated the cases when one can establish the uniform estimate for the Gauss curvature flow. In this paper, we study the $L_p$ dual Minkowski problem, an extension of the dual Minkowski problem. We deal with some cases in which there is no uniform estimate for the Gauss curvature flow. We adopt the topological method from ^[13] to find a special initial condition such that the Gauss curvature flow converges to a solution of the $L_p$ dual Minkowski problem.

In this paper, we use the Sobolev type inequality in Wang et al. (Moser–Trudinger inequality for the complex Monge–Ampère equation, arXiv: 2003.06056v1 (2020)) to establish the uniform estimate and the Hölder continuity for solutions to the complex Monge–Ampère equation with the right-hand side in $L^p$ for any given $p>1$. Our proof uses various PDE techniques but not the pluripotential theory.