The $L_p$ Minkowski problems on affine dual quermassintegrals

In this paper, {we extend the affine dual curvature measures to the $L_p$ setting and solve the existence part of the corresponding Minkowski problem for non-symmetric discrete measures when $p>1$ and for symmetric measures when $p\ge 0$.} When $p=0$, the $L_0$ Minkowski problem is the affine dual Minkowski problem, which is introduced and solved by Cai-Leng-Wu-Xi in [6].