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In this paper, we establish the singular Trudinger–Moser inequality withnorm in a bounded domainand the existence of extremal functions by blow-up analysis. As our first main result, we prove that for ,, and, the singular Trudinger–Moser inequalityholds, whereis the first eigenvalue . Furthermore, we prove the existence of extremals for the singular Trudinger–Moser inequality withnorm. Our results improve that of J. Zhu (Zhu J. Improved Moser–Trudinger inequality involvingnorm in  n  dimensions. Adv Nonlinear Stud 2014;14:273–293) into the singular case.

Singular Trudinger–Moser inequality with L p norm and the existence of extremals