By the two invariances from part 1, normalize a minimizing sequence $(u_n)$ so that
$$
\|u_n\|_2=\|\nabla u_n\|_2=1.
$$
Replacing $u_n$ by $|u_n|$ and then by its <symmetric decreasing rearrangement> preserves its $L^2$ and $L^p$ norms and does not increase the gradient norm. A harmless dilation restores the normalization, so we may take the sequence nonnegative, radial, and radially decreasing.

The sequence has a weakly convergent subsequence in $H^1$. Radial compactness and the fixed scale give strong convergence in $L^p$; in particular the limit is nonzero because $\|u_n\|_p^p\to I^{-1}$. <Weak lower semicontinuity> of the $L^2$ and gradient norms then shows that the limit attains the infimum. This is the <existence of a Weinstein-functional minimizer>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 154 / 2 / 2 / Solution

Normalize a minimizing sequence so that its $L^2$ norm and gradient norm are fixed. <Symmetric decreasing rearrangement> does not increase the gradient norm and preserves every $L^p$ norm, while radial compactness prevents translation loss. A weakly convergent subsequence therefore converges strongly in the nonlinear $L^{2+4/d}$ norm and yields a nonnegative minimizer.

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