If $U\subseteq \mathbb{R}^n$ is a "nice" bounded domain, then we know that the embedding $H_0^1(U) \hookrightarrow L^2(U)$ is compact.
I want to show that the the embedding $H_0^1(U) \hookrightarrow L^2(U)$ is compact still holds for the unbounded domain $U=\{(x,y) : 0<y<e^{-x^2}\}$. If so, how can one go about proving it?
I appreciate any input!