A sharp inequality for Sobolev functions

Abstract

Let $N\geq 5$, $a>0$, $\Omega$ be a smooth bounded domain in $\mathbb{R}^{N}$, $2^*=\frac{2N}{N-2}$, $2^\#=\frac{2(N-1)}{N-2}$ and $||u||^2=|\nabla u|_{2}^2+a|u|_{2}^2$. We prove there exists an $\alpha_{0}>0$ such that, for all $u\in H^1(\Omega)\setminus\{0\}$, $$\frac{S}{2^{\frac 2N}}\leq\frac{||u||^2}{|u|_{2^*}^2}\left(1+\alpha_{0}\frac{|u|_{2^\#}^{2^\#}}{||u||\cdot|u|_{2^*}^{2^*/2}}\right).$$ This inequality implies Cherrier's inequality.