Capillary $L_p$-curvature problem

We prove a gradient estimate for a class of capillary curvature equations in the half-space. As an application, we prove the existence of an even, smooth, strictly convex solution to the even capillary $L_p$-curvature problem for all $1<p<k+1$ and all contact angles $\theta \in (0,\pi \mathrm{/}2)$.