If
$$ p = \frac{qN}{q+N}, $$
then
$$
q = \frac{pN}{N-p}
$$
Therefore, since, $u$ is compactly supported in $\Omega$, by the sharp Sobolev inequality of Aubin and Talenti (see Talenti's paper) and the Holder inequality,
$$
\|u\|_{L^q(\Omega)} \le C(N,p)\|\nabla u\|_{L^p(\Omega)} \le
C(N,p)|\Omega|^{\frac{1}{p}-\frac{1}{N}}\|\nabla u\|_{L^N(\Omega)}
= C(N,p)|\Omega|^{\frac{1}{q}}\|\nabla u\|_{L^N(\Omega)}, $$
where
$$
C(N,p) = \pi^{-\frac{1}{2}}N^{-1/p}\left(\frac{p-1}{N-p}\right)^{1-1/p}
\left\{\frac{\Gamma(1+N/2)\Gamma(N)}{\Gamma(N/p)\Gamma(1+N-N/p)}\right\}^{1/N}
$$
It is now straightforward to check that if $q \rightarrow \infty$, then $p\rightarrow N$ and $C(N,p) \sim q^{1-1/N}$.

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