  In this paper, we study stability problem of anisotropic capillary
hypersurfaces in an Euclidean half-space. We prove that any compact immersed
anisotropic capillary constant anisotropic mean curvature hypersurface in the
half-space is weakly stable if and only if it is a truncated Wulff shape. On
the other hand, we prove a Bernstein-type theorem for stable anisotropic
capillary minimal surfaces in the three dimensional half-space under Euclidean
area growth assumption.
