The main question is: is the following expression true? $$C^l \subset W^{l,p} \subset L^p$$
To expand: from what I know Sobolev space is a way to weaken the differentiability requirements for a function, being the space of the functions which are p-integrable and have p-integrable weak derivatives up to the l-th order. This seems a stronger requirement than just being $L^p$ but a softer one than being $C^l$, therefore the above claim, but I'm afraid there could be some pathological examples I'm not thinking of.
You can always construct a smooth function which is not in $L^p(\Omega)$ ( hence not in $W^{l,p}(\Omega)$ ) even for bounded $\Omega$ with regular boundary, just take a function which blow up ( enough ) near the boundary. But you can always approximate functions in $W^{l,p}(\Omega)$ by smooth functions.