The capillary $L_p$ dual Christoffel-Minkowski problem for $1<p<q\le k+1$ with $1\le k\le n$

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 even, smooth, strictly convex solution in the parameter range $1<p<q\le k+1$, where $1\le k\le n$.