The complex Monge-Ampère equation for complex homogeneous functions in \({\mathbb C}^n\) (Q2724135)

It is shown that the equation \((dd^cu)^n= g d\lambda\) with \(g\) a smooth (outside the origin), nonnegative, complex homogeneous function of order \(n(\alpha- 2)\), \(0<\alpha< {1\over n-1}\), and \(\lambda\) the Lebesgue measure in \(\mathbb{C}^n\), admits a smooth (outside the origin) plurisubharmonic solution, homogeneous of order \(\alpha\). In addition, if \(\alpha<{1\over n}\) and \(g\) is locally bounded, the solution is locally bounded. When some symmetries of the function \(g\) are assumed, the restrictions on the homogeneity order are weakened.