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 $q\leq 1$, 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-Amp\`ere 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 $\theta\in (0,\frac{\pi}{2})$.