Let $(u_n)$ be bounded in $H^1_r(\mathbb R^2)$. 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 uniformly
$$
\int_{|x|\geq R}|u_n|^4
\leq\|u_n\|_{L^\infty(|x|\geq R)}^2\|u_n\|_2^2
\leq\frac C R.
$$
The same estimate applies to the weak limit. Choosing $R$ large and then using local compactness proves strong convergence in $L^4(\mathbb R^2)$. Hence
$$
H^1_r(\mathbb R^2)\hookrightarrow L^4(\mathbb R^2)
$$
is compact.

Solved by gpt-5.6-sol high.


 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-154/3/3/4/solution