Complex $m$-Hessian type equations in weighted energy classes of $m$-subharmonic functions with given boundary value

TL;DR

This paper investigates the existence of solutions to complex m-Hessian equations within weighted energy classes of m-subharmonic functions, focusing on boundary value problems and subsolution conditions.

Contribution

It establishes conditions for the existence of solutions to complex m-Hessian equations in weighted energy classes with given boundary data, extending previous results.

Findings

Existence of solutions under subsolution conditions

Solutions characterized within weighted energy classes

Boundary value problems addressed for m-Hessian equations

Abstract

In this paper, we concern with the existence of solutions of the complex $m-$Hessian type equation $-\chi(u)H_{m}(u)=\mu$ in the class $\mathcal{E}_{m,\chi}(f,\Omega)$ if there exists subsolution in this class, where the given boundary value $f\in\mathcal{N}_m(\Omega)\cap MSH_m(\Omega).$