  In this paper, combining the $p$-capacity for $p\in (1, n)$ with the Orlicz
addition of convex domains, we develop the $p$-capacitary
Orlicz-Brunn-Minkowski theory. In particular, the Orlicz $L_{\phi}$ mixed
$p$-capacity of two convex domains is introduced and its geometric
interpretation is obtained by the $p$-capacitary Orlicz-Hadamard variational
formula. The $p$-capacitary Orlicz-Brunn-Minkowski and Orlicz-Minkowski
inequalities are established, and the equivalence of these two inequalities are
discussed as well. The $p$-capacitary Orlicz-Minkowski problem is proposed and
solved under some mild conditions on the involving functions and measures. In
particular, we provide the solutions for the normalized $p$-capacitary $L_q$
Minkowski problems with $q>1$ for both discrete and general measures.
