Corollary 12.10.132C73D81273C0

Proof. By corollary 12.10.12, $f$ agrees almost everywhere with a smooth function, so it belongs to $L^ p$ for every finite $p$. By proposition 15.28.7, the constant $c$ for which $\mathrm{MA}_\omega (v)=cf\, \mathrm{d}V$ has a bounded solution is $b_\omega (f)$. Thus proposition 15.20.1, with the trivial symmetry group, shows that the normalized family $u_\lambda -\sup _ Xu_\lambda $ is uniformly precompact as $\lambda \downarrow 0$ and that every cluster limit is a normalized solution with this constant. Each cluster limit is therefore a normalized threshold subsolution and equals $u$ by corollary 12.10.12. If the whole family did not converge uniformly, a sequence staying at a fixed positive uniform distance from $u$ would have a uniformly convergent subsequence, a contradiction. $\square$