Subsolution theorem in weighted energy classes of $m$-subharmonic functions with given boundary value

TL;DR

This paper extends the subsolution theorem for weighted complex m-Hessian equations to broader classes of boundary values, establishing existence results based on the presence of subsolutions.

Contribution

It generalizes previous results by proving the subsolution theorem in weighted energy classes with more general boundary conditions.

Findings

Existence of solutions under subsolution conditions.

Extension of subsolution theorem to broader classes.

Generalization of boundary value conditions.

Abstract

In this paper, we concern with the existence of solutions of the weighted complex $m$-Hessian 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 $f\in\mathcal{E}_m(\Omega)\cap MSH_m(\Om).$ This is a generalization of the result in the paper \cite{PDtaiwan} where we proved that the subsolution theorem is true in the class $\mathcal{E}_{m,\chi}(f,\Omega)$ in the case when the given boundary value $f\in\mathcal{N}_m(\Omega)\cap MSH_m(\Om).$