Corollary 10.31.56.B2E2FCD34986 Let $0\leq f\in L^ p(X,\mathrm{d}V)$, with $p>1$ and $\int _ Xf\, \mathrm{d}V>0$. For $\lambda >0$, the solution in theorem 10.31.44 of
\[ \mathrm{MA}_\theta (u)=\mathrm{e}^{\lambda u}f\, \mathrm{d}V \]
satisfies $u\sim \varphi $. Every normalized solution in $\mathcal E_\varphi $ of
\[ \mathrm{MA}_\theta (u)=cf\, \mathrm{d}V,\qquad c>0, \]
also satisfies $u\sim \varphi $. In particular, both existence statements hold in the ordinary singularity type of $\varphi $.
Proof.
Lemma 5.26.2, applied in a Hermitian background dominating $\theta $, gives uniform moments of every finite order against $\mathrm{d}V$ for normalized $\theta $-plurisubharmonic functions. Hölder's inequality therefore implies, for every finite $m$,
\[ \sup _{\substack {v\in \mathrm{PSH}(X,\theta )\\ \sup _ Xv=0}} \int _ X(-v)^ m f\, \mathrm{d}V<\infty . \]
Choose $m>2n+1$. For the solution of the exponential equation, set $S=\sup _ Xu$ and $v=u-S$. Then $v\in \mathcal E_\varphi $, $\sup _ Xv=0$, and
\[ \mu _ v=\mathrm{e}^{\lambda u}f\, \mathrm{d}V \leq \mathrm{e}^{\lambda S}f\, \mathrm{d}V. \]
The preceding proposition, applied to this fixed dominating measure, shows that $v$, and hence $u$, has the ordinary type of $\varphi $. For the normalized equation, apply the proposition to the measure $cf\, \mathrm{d}V$.
$\square$
Pome
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