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 $\lambda>0$. So if $u$ maximizes both $\|u\|_p$ and $\|u\|_q$, then $u^p=u^q$ a.e., which is only possible if $p=q$.

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 $p=q$.