A sharp inequality for Sobolev functions

Abstract

Let $N\ge 5$, $a>0$, $\Omega$ be a smooth bounded domain in ${\mathbb{R}}^N$, $2^{\ast }=\frac{2N}{N-2}$, $2^{\#}=\frac{2(N-1)}{N-2}$ and $||u|{|}^2=|\nabla u{|}_2^2+a|u{|}_2^2$. We prove there exists an ${\alpha }_0>0$ such that, for all $u\in H^1(\Omega )\setminus \{0\}$, $$\frac{S}{2^{\frac{2}{N}}}\le \frac{||u|{|}^2}{|u{|}_{2^{\ast }}^2}\left(1+{\alpha }_0\frac{|u{|}_{2^{\#}}^{2^{\#}}}{||u||\cdot |u{|}_{2^{\ast }}^{2^{\ast }/2}}\right).$$ This inequality implies Cherrier's inequality.

4 pages

References in corpus (1)

• Existence and nonexistence of least energy solutions of the Neumann problem for a semilinear elliptic equation with critical Sobolev exponent and a critical lower-order perturbation

Cited by in corpus (1)

• Existence and nonexistence of least energy solutions of the Neumann problem for a semilinear elliptic equation with critical Sobolev exponent and a critical lower-order perturbation