Let $\Omega$ be a domain with a nice boundary (i.e., smoothness of the boundary shall not be central to my question).
Now it well-known that the embedding $\iota \colon W^{1,p}(\Omega) \to L^q(\Omega)$ is compact whenever $1 \le p < n$ and $q < p^* = \frac{np}{n-p}$.
In the limit case $q = p^*$, the embedding is not generally compact.
My question is: Is it in some sense better than continuous?
In other words: Say we have a continuous map $T \colon L^q(\Omega) \to W^{1,p}(\Omega)$. If $q < p^*$, then $\iota$ is compact and so are the composition of $T$ and $\iota$ (regardless of the order). Consequently, if said composition is a self-map on a non-empty bounded closed convex set, then it has a fixed point by Schauder's theorem. In the case $q = p^*$, if $\iota$ were still, e.g., condensing, we would again have a fixed point by Darbo's theorem...