A family of sharp inequalities for Sobolev functions

Abstract

Let $N\ge 5$, $\Omega$ be a smooth bounded domain in ${\mathbb{R}}^N$, $2^{\ast }=\frac{2N}{N-2}$, $a>0$, $S=\inf \left\{{\int }_{{\mathbb{R}}^N}|\nabla u{|}^2|u\in L^{2^{\ast }}({\mathbb{R}}^N),\nabla u\in L^2({\mathbb{R}}^N),{\int }_{{\mathbb{R}}^N}|u{|}^{2^{\ast }}=1\right\}$ and $||u|{|}^2=|\nabla u{|}_2^2+a|u{|}_2^2$. We define $2^{♭}=\frac{2N}{N-1}$, $2^{\#}=\frac{2(N-1)}{N-2}$ and consider $q$ such that $2^{♭}\le q\le 2^{\#}$. We also define $s=2-N+\frac{q}{2^{\ast }-q}$ and $t=\frac{2}{N-2}\cdot \frac{1}{2^{\ast }-q}$. We prove that there exists an ${\alpha }_0(q,a,\Omega )>0$ such that, for all $u\in H^1(\Omega )\setminus \{0\}$, $$\frac{S}{2^{\frac{2}{N}}}|u{|}_{2^{\ast }}^2\le ||u|{|}^2+{\alpha }_0{\left(\frac{||u||}{|u{|}_{2^{\ast }}^{2^{\ast }/2}}\right)}^s|u{|}_q^{qt},(I{)}_q$$ where the norms are over $\Omega$. Inequality $(I{)}_{2^{♭}}$ is due to M. Zhu.

arXiv comment25 pages. arXiv admin note: text overlap with arXiv:1407.6232