A family of sharp inequalities for Sobolev functions

Abstract

Let $N\geq 5$, $Ω$ be a smooth bounded domain in $\mathbb{R}^{N}$, ${2^*}=\frac{2N}{N-2}$, $a>0$, $S=\inf\left\{\left. \int_{\mathbb{R}^{N}}|\nabla u|^2\,\right|\,u\in L^{2^*}(\mathbb{R}^{N}), \nabla u\in L^2(\mathbb{R}^{N}), \int_{\mathbb{R}^{N}}|u|^{2^*}=1 \right\}$ and $||u||^2=|\nabla u|_{2}^2+a|u|_{2}^2$. We define ${2^\flat}= \frac{2N}{N-1}$, ${2^\#}=\frac{2(N-1)}{N-2}$ and consider $q$ such that ${2^\flat}\leq q\leq{2^\#}$. We also define $s=2-N+\frac{q}{{2^*}-q}$ and $t=\frac{2}{N-2}\cdot \frac{1}{{2^*}-q}$. We prove that there exists an $α_{0}(q,a,Ω)>0$ such that, for all $u\in H^1(Ω)\setminus\{0\}$, $$\frac{S}{2^{\frac 2N}}{|u|_{2^*}^2}\leq||u||^2+α_{0} \left(\frac{||u||}{|u|_{2^*}^{2^*/2}}\right)^s|u|_{q}^{qt},\qquad{(I)_{q}}$$ where the norms are over $Ω$. Inequality $(I)_{2^\flat}$ is due to M. Zhu.