I am looking for some references (if there are) of continuous or compact injections satisfied by the Sobolev-type spaces $$ W_{1,p,q}(\Omega) := \{ v \in L^p(\Omega): \nabla v \in [L^q(\Omega)]^n \}, $$ where $\Omega \subseteq \mathbb{R}^n$ is an open bounded set.
I am especially interested in the possible compactness of the injection $W_{1,p,q}(\Omega) \hookrightarrow L^p(\Omega)$.
Let me assume that $\Omega$ allows for the standard Sobolev embedding theorems to hold.
Case 1: $p \le q$. Then $W_{1,p,q} \hookrightarrow W^{1,p}$, which is compactly embedded into $L^p$.
Case 2: $p>q$. Then $W_{1,p,q} \hookrightarrow W^{1,q}$. If this is compactly embedded into $L^p$, we are done. If not, then $W^{1,q}$ is compactly embedded into $L^q$. The compactness of the embedding $W_{1,p,q} \hookrightarrow L^q$ and the continuity of $W_{1,p,q} \hookrightarrow L^p$ imply (by Hoelder inequality) the compactness of $W_{1,p,q} \hookrightarrow L^r$ for all $r<p$.
Here is an example, that it is impossible to get $r=p$ in the latter case: $n=1$, $p=\infty$, $q=1$, $\Omega=(-1,1)$. Set $u_n(x)=\max(-1,\min(nx,+1))$ and $u(x):=sign(x)$. Then $(u_n)$ is bounded in $L^\infty$, $(u_n')$ is bounded in $L^1$, $u_n\to u$ in $L^r$ for all $r<\infty$ but $u_n\not\to u$ in $L^\infty$.