Consider the standard Sobolev space $H\equiv H_0^1(\Omega)$, where $\Omega\subset \mathbb{R}^N$ is a bounded smooth domain. Let us consider on $H$ the norm $$\|u\|^2=\int |\nabla u|^2,$$
and define $S\subset H$ by $$S=\{u\in H_0^1(\Omega): \|u\|=1\}.$$
Let $1<q<p<2^\star$ where $2^\star$ is the critical Sobolev exponent. We know that $$\tag{1}0<C_p=\inf_{u\in S}\frac{1}{ \|u\|_p},$$
and $$\tag{2}0<C_q=\inf_{u\in S}\frac{1}{\|u\|_q}.$$
So, my question is the following. Let $C_{p,q}$ be defined by $$C_{p,q}=\inf_{u\in S}\frac{1}{\|u\|_q\|u\|_p}.$$
Is it true that $C_{p,q}=C_pC_q$?
My thoughts on the problem: The answer seems to be true to me, because of the inequality $\|u\|_{q}\le C\|u\|_p$, so it seems that when one maximizes the norm $\|u\|_q$ over $S$, the norm $\|u\|_p$ is also maximized over $S$, however, the argument is not that simple.
Any idea is aprreciated.