Poincaré’s inequality for a bounded open set in Brezis' book
Solution 1:
For any $1\le p_1<N$ we have the embedding $$ W^{1,N}_0 (\Omega) \to W^{1,p_1}_0 (\Omega)$$ since $\Omega$ is bounded. Now choose $p_1 <N$ so that
$$ p_1^* = \frac{p_1N}{N-p_1}> N. $$
Then there is a compact embedding
$$ W^{1,N}_0 (\Omega) \to W^{1,p_1} (\Omega) \to L^N (\Omega).$$
(the second $\to$ is compact). Then for all $u\in W^{1,N}_0(\Omega)$,
$$ \| u\|_{L^N} \le C \|u\|_{L^{p_1^*}} \le C \| \nabla u\|_{L^{p_1}} \le C \| \nabla u\|_{L^N}.$$