The solvability of the $L_p$-dual mixed Euclidean-Gauss Minkowski problem

This paper investigates the $L_p$-dual mixed Euclidean-Gauss Minkowski problem associated with the $L_p$-dual mixed Euclidean-Gaussian curvature measure. The solvability of this problem is equivalent to that of a class of Monge-Ampère type equations on the unit sphere. Under smooth assumptions, We establish the existence of solutions for $p<1$, $q<0$, and $q<p$ and uniqueness of solutions for $p\ge q$ to the aforementioned Monge-Ampère type equations, respectively. Furthermore, we obtain a complete solution concerning the existence part of this problem when $p<1$ and $q<0$.