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The main aim of this article is to study non-singular version of Moser–Trudinger and Adams–Moser–Trudinger inequalities and the singular version of Moser–Trudinger equality in the Cartesian product of Sobolev spaces. As an application of these inequalities, we study a system of Kirchhoff equations with exponential non-linearity of Choquard type.

Adams–Moser–Trudinger inequality in the Cartesian product of Sobolev spaces and its applications