On some weighted energy classes of plurisubharmonic functions

In this paper we study the relation between the weighted energy class $\mathcal{E}_{\chi}$ introduced by S. Benelkouchi, V. Guedj and A. Zeriahi recently with the classes $\mathcal{E}$ and $\mathcal{N}$ studied by Cegrell. Moreover, we establish a generalized comparison principle for the operator $\text{M}_{\chi}$ and, as an application, we prove a slight version of existence of solutions of Monge-Amp\`ere type equation in the class $\mathcal{E}_{\chi}(H,\O)$.