 Solution

Let $(u_n)$ be bounded in $H_r^1({\mathbb{R}}^2)$. After passing to a subsequence it converges weakly in $H^1$, while the Rellich-Kondrachov compactness theorem gives strong $L^4$ convergence on every bounded ball. The Radial Sobolev inequality gives uniformlyThe same estimate applies to the weak limit. Choosing $R$ large and then using local compactness proves strong convergence in $L^4({\mathbb{R}}^2)$. Henceis compact.