The ordinary Sobolev embedding states that, if $\Omega\subset \mathbb R^n$ is a bounded Lipschitz domain, then for $kp>n$ we have an embedding $W^{k,p}(\Omega) \subset C^0(\Omega)$; while for $kp<n$ we have $W^{k,p}(\Omega) \subset L^q(\Omega)$ for $q< \frac{np}{n-kp}$.
Question: Is there any version of the Sobolev embedding theorem which is with respect to $W^{k,p}_0(\Omega)$ instead of $W^{k,p}(\Omega)$? What if we assume $\Omega$ is unbounded?
For my interest, is there a version that states $W^{1,p}(\mathbb R) \subset C^0(\mathbb R)$?
Notation: Denote by $C^\infty_0$ the set of all real-valued smooth functions $f$ on $\mathbb R$ such that $\lim_{x\to \pm\infty} f(x)=0$. Let $W^{1,p}_0$ be the completion of $C^\infty_0$ with respect to the Sobolev norm $||\cdot||_{W^{1,p}}$
For $W_0^{k,p}(\Omega)$ with $\Omega \subset \mathbb R^n$ open, yes. The idea is to establish the relevant inequalities for $u \in C_c^{\infty}(\Omega)$ (which is dense in $W_0^{1,p}(\Omega)$ as you define it by mollification) and extend by density, which is probably what you learnt in the ordinary version of the theorem minus the extra step of extending it via extension operators.
This is done for example in Evans' PDE book, where he proves the Gagliardo-Nirenburg-Sobolev inequality and Morrey's inequality in $C_c^1(\mathbb R^n).$ The same proof however, can also be used to establish the result in $C_c^1(\Omega)$ for $\Omega \subset \mathbb R^n$ open.