On a class of N-dimensional anisotropic Sobolev inequalities

Abstract

In this paper, we study the smallest constant α in the anisotropic Sobolev inequality of the form ∥u∥pp≤α∥u∥22(2N-1)+(3-2N)p2∥ux∥2N(p-2)2∏k=1N-1∥Dx-1∂yku∥2p-22 and the smallest constant β in the inequality ∥u∥p∗p∗≤β∥ux∥22N2N-3∏k=1N-1∥Dx-1∂yku∥222N-3, where V:=(x,y1,…,yN-1)∈RN with N≥3 and 2<p<p∗=2(2N-1)2N-3 . These constants are characterized by variational methods and scaling techniques. The techniques used here seem to have independent interests.