Let $\Omega\subset \mathbb{R}^N$ and $H_0^1(\Omega)$ the standard Sobolev space. Assume that $1<q<p<2^\star$ and $$\mathcal{S}=\{u\in H_0^1(\Omega):\ \|u\|=1\}.$$

Define $C_q,C_p$ by $$C_q=\inf_{u\in \mathcal{S}}\frac{1}{\|u\|_q},$$

and $$C_p=\inf_{u\in \mathcal{S}}\frac{1}{\|u\|_p}.$$

Once $q<p<2^\star$, we can assume the existence of $u_q,u_p\in \mathcal{S}$ such that $$C_q=\frac{1}{\|u_q\|_q}, C_p=\frac{1}{\|u_p\|}.$$

My question is the following:

Can we have $u_q=u_p$, or equivalently, does the maximum of the function $\|u\|_q\|u\|_p$ over $\mathcal{S}$ is attained for any function, which maximizes both $\|u\|_q$ and $\|u\|_p$ in the same time?

My guess by now is that this is not true, however, I have no idea to how to prove it. Any reference on this matter is appreciated.

I have asked the same question here.