The $L_p$ dual Minkowski problem for capillary hypersurfaces

In this paper, we consider the $L_p$ dual Minkowski problem for capillary hypersurfaces for $p>q$ and $q\le 1$, which aims to find a capillary convex body with a prescribed capillary $(p,q)$-th dual curvature measure in the Euclidean half-space. We reduce it to a Monge-Ampère type equation with a Robin boundary condition on the unit spherical cap, we prove that there exists a unique smooth solution that solves this problem provided $\theta \in (0,\frac{\pi }{2})$.