Abstract
In this paper, we establish the singular Trudinger–Moser inequality with norm in a bounded domain and the existence of extremal functions by blow-up analysis. As our first main result, we prove that for , , and , the singular Trudinger–Moser inequality holds, where is the first eigenvalue . Furthermore, we prove the existence of extremals for the singular Trudinger–Moser inequality with norm. Our results improve that of J. Zhu (Zhu J. Improved Moser–Trudinger inequality involving norm in n dimensions. Adv Nonlinear Stud 2014;14:273–293) into the singular case.
Acknowledgments
The authors wish to sincerely thank the referees for their very careful reading of the paper and for many helpful comments which have improved the exposition of the paper.
Disclosure statement
No potential conflict of interest was reported by the author(s).