On a Monge–Ampère type equation in the Cegrell class $\mathcal{E}_{\chi}$

Volume 99 / 2010

Rafa/l Czyż Annales Polonici Mathematici 99 (2010), 89-97 MSC: Primary 32W20; Secondary 32U15. DOI: 10.4064/ap99-1-8


Let $\Omega$ be a bounded hyperconvex domain in ${\mathbb C}{n}$ and let $\mu$ be a positive and finite measure which vanishes on all pluripolar subsets of $\Omega$. We prove that for every continuous and strictly increasing function $\chi:(-\infty,0) \to (-\infty,0)$ there exists a negative plurisubharmonic function $u$ which solves the Monge–Ampère type equation $$ -\chi(u)(dd^cu)^n = d\mu. $$ Under some additional assumption the solution $u$ is uniquely determined.


  • Rafa/l CzyżInstitute of Mathematics
    Jagiellonian University
    Łojasiewicza 6
    30-348 Kraków, Poland

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