Equations of complex Monge-Ampère type for arbitrary measures and applications (Q2807999)

Let \(\Omega\subset \mathbb C^n\) be a bounded hyperconvex domain, let \(\mathcal E(\Omega)\) be the maximal class of negative plurisubharmonic (PSH) functions for which the complex Monge-Ampère operator \((dd^c\cdot)^n\) is well defined and let \(\mathcal N(\Omega)\subset \mathcal E(\Omega)\) be set of functions with smallest maximal plurisubharmonic majorant equal to zero. Let \(\mathcal N(\Omega, f)\) be the corresponding class with generalized boundary values \(f\in \mathcal E(\Omega)\), i.e., \(u\in \mathcal N(\Omega, f)\) if there exists \(v\in \mathcal N(\Omega)\) such that \(f\geq u\geq f+v\).NEWLINENEWLINEThe aim of this paper is to prove the existence of a solution to the complex Monge-Ampère type equation in \(\Omega\subset \mathbb C^n\): NEWLINE\[NEWLINE (dd^cu)^n=F(u,\cdot)d\mu, NEWLINE\]NEWLINE where \(\mu\) is a positive Borel measure, \(F:\mathbb R\times \Omega\to[0,\infty)\) is a measurable function and \(u\in \mathrm{PSH}(\Omega)\).NEWLINENEWLINEThe main result of this paper is the following theorem. Let \(\mu\) be a Borel measure in a bounded hyperconvex domain \(\Omega\) and let \(F(t,z)\) be a positive function such that {\parindent=0.6cm\begin{itemize}\item[(a)] for all \(z\) the function \(t\mapsto F(t,z)\) is continuous and nondecreasing, \item[(b)] for all \(t\) the function \(z\mapsto F(t,z)\) is locally \(\mu\)-integrable, and \item[(c)] there exists a subsolution \(w\in \mathcal N(\Omega)\) such that \((dd^cw)^n\geq F(w,\cdot)d\mu\). NEWLINENEWLINE\end{itemize}} Then for any maximal plurisubharmonic function \(f\in \mathcal E(\Omega)\) there exists a solution \(u\in \mathcal N(\Omega,f)\) to the equation \((dd^cu)^n=F(u,\cdot)d\mu\) such that \(u\geq w\).NEWLINENEWLINEThis theorem generalizes a recent result by Benelkourchi, proved for measures vanishing on pluripolar sets [\textit{S. Benelkourchi}, Ann. Pol. Math. 112, No. 3, 239--246 (2014; Zbl 1308.32044)].