Corollary 12.10.35.D0692BB5EDD8 For the density and potential of proposition 12.10.33,
\[ \int _ X f\log f\, \mathrm{d}V<+\infty \quad \Longleftrightarrow \quad \alpha <1-\frac1n. \]
For every real $r>0$,
\[ \int _ X(-u)^ r f\, \mathrm{d}V<+\infty \quad \Longleftrightarrow \quad \alpha r<n(1-\alpha ). \]
In particular, finite entropy of the density can coexist with an unbounded threshold solution of full mass and with the absence of a smooth semipositive maximizing test.
Proof.
Only a neighborhood of $o$ can contribute a divergence. There the preceding estimates give $f\, \mathrm{d}V\asymp L^{-1-n(1-\alpha )}\, \mathrm{d}L$, $\log f\sim nL$ and $-u\asymp L^\alpha $. Hence the entropy integral converges exactly when $n(1-\alpha )>1$, and the weighted integral exactly when $\alpha r<n(1-\alpha )$. In the equality cases both integrals are comparable to $\int ^\infty L^{-1}\, \mathrm{d}L$ and diverge. The negative part of $f\log f$ is bounded, hence integrable for the probability volume, so the entropy criterion is unambiguous.
$\square$
Pome
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