Counter example of rellich-kondrachov compact embedding theorem on unbounded domain

The Rellich-Kondrachov Compactness Theorem says that when $$U$$ is a bounded set with $$C^1$$ boundary then $$W^{1,p}(U)$$ is compactly embedded into $$L^{q}(U)$$ for every $$1 \leq q < p^{*}$$.

What if $$U$$ is unbounded, e.g. $$U=\mathbb{R}^n$$?

As Jake pointed out, the embedding fails in $$\mathbb{R}^{n}$$ because we can find a sequence of bumps bounded in $$W^{1,p}(\mathbb{R}^{n})$$ that go to infinity.

However, it turns out that there are unbounded domains $$U$$ with $$W^{1,p}(U)\subset \subset L^{p}(\Omega)$$ - you can find some of them in the book Sobolev Spaces by Adams.

In the more specific case of $$W^{1,p}_{0}(U)$$ for example, there are a few nice geometric conditions we can impose on unbounded $$U$$ so that the embedding is compact.

A necessary one is the following: $$U$$ must not contain infinitely many disjoint $$\epsilon$$-balls - otherwise, choose a sequence of bumps in those balls and apply the same argument as in $$\mathbb{R}^{n}$$ to contradict compactness.

Actually, in $$n=1$$ dimension, this is even sufficient and there is a nice elementary proof using only the fundamental theorem of calculus.

In higher dimensions, we need $$U$$ to be "sufficiently thin as we approach infinity" - what exactly that means you can read about in the book mentioned above.

Take a bump function $$f$$ with compact support. Define $$f_n$$ to be a translation of $$f$$ in some direction, say along the first coordinate axis. These clearly don't have a limit in $$L^q$$.