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

The $p$-capacitary Orlicz-Hadamard variational formula and Orlicz-Minkowski problems