Best constants and existence of maximizers for weighted Trudinger–Moser inequalities

Abstract

The main purpose of this paper is three-fold. First of all, we will establish a weighted Trudinger–Moser inequality on the entire space without requiring the functions under consideration being radially symmetric (see Theorem 1.1). We will also prove the existence of a maximizer of this sharp weighted inequality. Therefore, our result improves the earlier one where such type of inequality has only been proved for spherically symmetric functions by M. Ishiwata, M. Nakamura, H. Wadade in (Ann Inst H Poincare Anal Non Linaire 31(2):297–314, 2014) (except in the case \(s\not =0\)). Second, we will prove stronger weighted Trudinger–Moser inequalities for functions which are not required to be radially symmetric (see Theorems 1.2 and 1.3). Finally, the existence of a maximizer of these stronger inequalities is proved in Theorem 1.4.