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

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 ${\mathcal{E}}_{m,\chi }(f,\mathrm{\Omega })$ if there exists subsolution in this class, where the given boundary value $f\in {\mathcal{E}}_m(\mathrm{\Omega })\cap MS{H}_m(\text{Om})\mathrm{.}$ This is a generalization of the result in the paper \cite{PDtaiwan} where we proved that the subsolution theorem is true in the class ${\mathcal{E}}_{m,\chi }(f,\mathrm{\Omega })$ in the case when the given boundary value $f\in {\mathcal{N}}_m(\mathrm{\Omega })\cap MS{H}_m(\text{Om})\mathrm{.}$