Sobolev inequality in $W_0^{1,p}$ If $\Omega \subseteq \mathbb{R}^N$ is an open bounded domain and $1<p<N$, then the classical Sobolev Inequality:
$$\| u\|_{p^*,\Omega} \leq C\ \| \nabla u\|_{p,\Omega}$$
holds with $C=C(p,N,\Omega)>0$ and $p^*:= Np/(N-p)$ for any $u\in W_0^{1,p}(\Omega)$.
What about the case $p\geq N$?
May I take the $L^\infty$-norm in the LHside? 
If I remember correctly, in general I cannot get the inequality with $\| \cdot \|_\infty$, for there are counterexemples of unbounded Sobolev functions... But, what if I know "a priori" that $u\in L^\infty(\Omega) \cap W_0^{1,p}(\Omega)$?
Any reference? (Adams-Fournier? Brezis?)
Thanks in advance.
 A: For the case $p=N$ take a look here. There you will find all you want. Note that the function $f$ defined by me is a counter example for what you want, also, even if you ask $u\in L^\infty$, you dont get what you want. Take a look in the answer and you will have all the explanations you need. 
When $p>N$ and you have some regularity in the boundary, then your functions are continuous, even Holder continuous. I suggest you to take a look in any good book about Sobolev Spaces. For example the book ok Leoni is a good one, there you will find all you need. 
A: When $\Omega\subset%
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\mathbb{R}
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^{n}$, $n\geq2$ is a bounded domain, we have by the Sobolev embedding theorems
that $W_{0}^{k,p}\left(  \Omega\right)  \subset L^{q}\left(  \Omega\right)
,~1\leq q\leq\frac{np}{n-kp},~kp<n$ and that $W_{0}^{k,\frac{n}{k}}\left(
\Omega\right)  \subset L^{q}\left(  \Omega\right)  ,~1\leq q<\infty$. However,
there are counterexamples to the embedding $W_{0}^{k,\frac{n}{k}}\left(
\Omega\right)  \subset L^{\infty}\left(  \Omega\right)  $. In this case, it
was proposed independently by Yudovich, Pohozaev and
Trudinger that $W_{0}^{1,n}\left(  \Omega\right)  \subset
L_{\varphi_{n}}\left(  \Omega\right)  $ where $L_{\varphi_{n}}\left(
\Omega\right)  $ is the Orlicz space associated with the Young function
$\varphi_{n}(t)=\exp\left(  \beta_{n} \left\vert t\right\vert ^{n/(n-1)}%
\right)  -1$ for some positive $\beta_{n}$. Moreover, Moser explored more in
this direction and further found the largest positive real number $\beta_{n}$.
In fact, in his 1971 paper [A sharp form of an inequality by N. Trudinger. Indiana Univ. Math. J. 20 (1970/71), 1077–1092.], J. Moser proved the following result: There exist sharp constant $\beta_{n}%
=n\omega_{n-1}^{\frac{1}{n-1}}$, where $\omega_{n-1}$ is the
area of the surface of the unit $n-$ball, such that 
$$
\frac{1}{\left\vert \Omega\right\vert }\int_{\Omega}\exp\left(  \beta
\left\vert u\right\vert ^{\frac{n}{n-1}}\right)  dx\leq c_{0}%
$$
for any $\beta\leq\beta_{n}$, any $u\in W_{0}^{1,n}\left(
\Omega\right)  $ with $\int_{\Omega}\left\vert \nabla u\right\vert
^{n}dx\leq1$. This constant $\beta_{n}$ is sharp in the sense
that if $\beta>\beta_{n}$, then the above inequality can no longer
hold with some $c_{0}$ independent of $u$.
The same result for $W_{0}^{k,p}\left(  \Omega\right)$ was proved by D. Adams in his paper [A sharp inequality of J. Moser for higher order derivatives. Ann. of Math. (2) 128 (1988), no. 2, 385–398.]
A: Sobolev inequality gives you that $W_0^{m,p}$ is embedded $L^{p^*}$, and by an interpolation argument, you can embed $W_0^{m,p}$ in $L^q$ for $p\leq q\leq p^*$ for $mp<N$, in your case, $m = 1$, $p<N$.
If you want $mp=N$, $W_0^{m,p}$ is embedded in $L^q$ with $1<p\leq q <\infty$. In your case $m=1$, $p=N$, but I am not sure what happens if $p>N$.
For a proof of these facts, see page 20 of 
http://people.bath.ac.uk/masgrb/Sobolev/notes.pdf
I am not sure if this answers your question. I am also only a beginner in this area.
