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In this paper, we study a Dirichlet type problem for the non-pluripolar complex Monge - Amp\`ere equation with prescribed singularity on a bounded domain of Cⁿ. We provide a local version for an existence and uniqueness theorem proved by Darvas, Di Nezza and Lu. Our work also extends a result of Ahag, Cegrell, Czyz and Pham.
Do et al. (Sun,) studied this question.