Hypersurfaces of Prescribed Mixed Weingarten Curvature

The primary objectives of this paper can be categorized into two main aspects. First, we investigate the existence of a closed hypersurface in Euclidean space with prescribed mixed quotient type Weingarten curvature. This result can be viewed as an improvement of the result of Chen-Tu-Xiang [12, Theorem 1.1]. Second, we study the homogeneous mixed Weingarten curvature problem and the convexity of the corresponding hypersurface.