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

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 ${\mathcal{E}}_{m,\chi }(f,\mathrm{\Omega })$ if there exists subsolution in this class, where the given boundary value $f\in {\mathcal{N}}_m(\mathrm{\Omega })\cap MS{H}_m(\mathrm{\Omega })\mathrm{.}$