25.9 Uniqueness and selectionAFDCC356D816

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When the exponent is zero, uniqueness of the normalization constant leaves open the separate question of uniqueness of the potential. Degeneration of the density affects comparison, whereas the family of equations with positive exponent can select a particular solution of the critical equation. The argument by concentration of mass for densities restricted to a set, around (14.25.6), follows Lu, Phung and Tô [proof of Theorem 3.7 and Remark 3.8, LPT20]. The exterior-ball argument for (14.22.11) instead propagates the maximum, following Bardi and Goffi [Proposition 2.4 and Theorem 2.5 in the cited preprint, BG18].

The uniqueness theorem for bounded potentials, theorem 14.24.1, concerns a continuous nonnegative density whose positivity set is connected and dense, and is proved with energy estimates on level sets and auxiliary test potentials. The mixed inequality and the passage to coefficient densities use the local calculus of section 5.21, and the continuity of bounded solutions is obtained in the book itself, by a comparison argument, in corollary 14.19.29. The smooth local Dirichlet problem needed for proposition 14.24.13 is solved by Guan's theorem [Theorem 1.1, pp. 687–688, Guan98], whose hypothesis is satisfied by a strictly plurisubharmonic quadratic subsolution. The estimate on the real contact set then gives a constant that depends only on the fixed geometry and the $L^2$ bound, and not on the height of the test density.

For measurable densities that are bounded above and bounded away from zero, uniqueness is derived in corollary 14.24.17 from the same energy argument on level sets. The classical counterpart is [Corollary 3.9 and Remark 3.13, KN19], where the locators refer to the authors' version indicated in the bibliography.