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](https://math.stackexchange.com/questions/1879542/infimum-of-frac1-u-q-u-p-over-the-sphere-in-h-01-omega).



Maximizing $\|u\|_q\|u\|_p$ over the unitary sphere in the Sobolev space $H_0^1(\Omega)$
Any maximizer $u\in\{ H^1\_0(\Omega): \|\nabla u\|\_2=1 \}$ of $\|u\|\_p$ is a nonconstant, nonnegative function solving $-\Delta u = \lambda u^{p-1}$, with $\lambda=\lambda\_p>0$. So if $u$ maximizes both $\|u\|\_p$ and $\|u\|\_q$, then $\lambda\_p u^p=\lambda\_q u^q$ a.e., which is only possible if $p=q$. 


