On the anisotropic Caffarelli–Kohn–Nirenberg type inequalities: Existence, symmetry breaking region and symmetry of extremal functions

In 2021, Li and Yan considered a more general and improved anisotropic version of the Caffarelli–Kohn–Nirenberg celebrated interpolation inequalities [Formula: see text] in dimensions [Formula: see text], where [Formula: see text] and [Formula: see text], and gave necessary and sufficient conditions for which to hold under natural assumptions on the parameters. In this paper, we study the Li–Yan inequality with [Formula: see text], [Formula: see text], and answer some fundamental questions concerning these inequalities such as the existence and symmetry breaking region of extremal functions, and their symmetry properties.