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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.

Brunn-Minkowski inequality, M-addition, Minkowski inequality, Minkowski problem, mixed p- capacity, Orlicz addition of convex bodies, Orlicz-Brunn-Minkowski theory, Orlicz Minkowski problem, p-capacity