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 ${\mathcal{E}}_{\chi }(H,\text{O})$.