The Minkowski problem of $p$-affine dual curvature measures

For $p\in (-\infty,0)\cup(0,1)$ and a convex body $K\subset\mathbb{R}^n$ with the origin in its interior, we construct the family of $p$-affine dual curvature measures $\mathcal{I}_p(K,\cdot)$ with respect to $K$. The affine-invariant measure $\widetilde{C}_{n-1}^{\mathrm{a}}(K, \cdot)$ given in the paper [9] is the limit case of $\mathcal{I}_p(K,\cdot)$ as $p\rightarrow 1^-$. The classical cone-volume measure is the limit case of the affine measures $2|p|\mathcal{I}_p(K,\cdot)/(n^2V(\mathrm{I}_pK))$ when $p\rightarrow 0$ and $V(K)=2$, where $\mathrm{I}_pK$ denotes the $L_p$ intersection body of $K$. The Minkowski problems for the $p$-affine dual curvature measures are proposed and studied. Specifically, we give a sufficient condition for the existence of a solution to the even Minkowski problem for $p$-affine dual curvature measure. Moreover, a necessary condition is given when $p\in (0,1)$. The smooth case of this Minkowski problem is equivalent to solving a new type of partial differential equations with respect to $p$-cosine transforms.