In this paper, we establish a weighted Trudinger-Moser type inequality with the full Sobolev norm constraint on the whole Euclidean space. Main tool is the singular Trudinger-Moser inequality on the whole space recently established by Adimurthi and Yang, and a transformation of functions. We also discuss the existence and non-existence of maximizers for the associated variational problem.
Van Hoang Nguyen. Futoshi Takahashi. "On a weighted Trudinger-Moser type inequality on the whole space and related maximizing problem." Differential Integral Equations 31 (11/12) 785 - 806, November/December 2018. https://doi.org/10.57262/die/1537840869