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 ${\parallel u\parallel }_p^p\le \alpha {\parallel u\parallel }_2^{\frac{2(2N-1)+(3-2N)p}{2}}{\parallel u_x\parallel }_2^{\frac{N(p-2)}{2}}\prod _{k=1}^{N-1}{\parallel D_x^{-1}{\partial }_{y_k}u\parallel }_2^{\frac{p-2}{2}}$ and the smallest constant β in the inequality ${\parallel u\parallel }_{p_{\ast }}^{p_{\ast }}\le \beta {\parallel u_x\parallel }_2^{\frac{2N}{2N-3}}\prod _{k=1}^{N-1}{\parallel D_x^{-1}{\partial }_{y_k}u\parallel }_2^{\frac{2}{2N-3}},$ where $V:=(x,y_1,\dots ,y_{N-1})\in R^N$ with $N\ge 3$ and $2<p<p_{\ast }=\frac{2(2N-1)}{2N-3}$ . These constants are characterized by variational methods and scaling techniques. The techniques used here seem to have independent interests.