The capillary $L_{p}$ dual Christoffel-Minkowski problem for $1<p<q\leq k+1$ with $1\leq k\leq n$

This paper is concerned with a capillary $L_p$ Christoffel--Minkowski-type problem involving the prescription of a class of $L_p$ geometric measures formed by the $k$-th capillary area measure and the $q$-th dual curvature measure. By establishing a gradient estimate, we prove the existence of an ev