 Solution

On the unit ball, part 1 controls the $L^2$ norm. Outside it, $|x{|}^{\alpha }\ge 1$, soHence $\Vert u{\Vert }_2\le C\Vert u{\Vert }_{\Sigma }$, proving that the embedding $\Sigma \hookrightarrow L^2$ is continuous.

For compactness, let $(u_n)$ be bounded in $\Sigma$. On each ball, it is bounded in $H^1$, so the Rellich-Kondrachov compactness theorem gives a subsequence convergent in local $L^2$. The tail estimateis uniform in $n$ and tends to zero as $R\to \infty$. A diagonal argument therefore gives convergence in all of $L^2$. This is the compact embedding of a confining-potential energy space.