On the embedding of a Sobolev space on a unbounded domain

Question

I have a question about a Sobolev space.

Let $D \subset \mathbb{R}^d$ be a connected open subset with smooth boundary. When $D$ is bounded, Relich's theorem states that $W^{1,2}(D)$ (Sobolev space on $D$ with Neumann boundary condition) is compactly embedded in $L^2(D)$. But I am interested in the case $D$ is unbouned.

Question

When $D=\{(x,y) \in \mathbb{R}^2: y>0 \}$, can we show that $W^{1,2}(D)$ is compactly embedded in $L^2(D)$ or $W^{1,2}_{0}(D)$ is compactly embedded in $L^2(D)$ ? ($W_{0}^{1,2}(D)$ is the Sobolev space on $D$ with Dirichlet boundary condition)

If you know related research, please let me know.

Answer 1

No. Let $\phi\in C^\infty_0(B_1)$ where $B_1$ is the ball of radius 1.

Let $\phi_n(x,y) = \phi(x,y - 10n)$. By construction $(\phi_n)$ is a sequence of functions in $C^\infty_0(D)$ with bounded $W^{1,2}$ norm, but has no converging subsequence in $L^2$.