A sharp inequality for Sobolev functions

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