Sharp Weighted Trudinger-Moser Inequalities with the Ln Norm in the Entire Space Rn and Existence of Their Extremal Functions

In this paper, we mainly concern with the sharp weighted Trudinger-Moser inequalities with the L-n norm on the whole space (See Theorem 1.1 and 1.3). Most proofs in the literature of existence of extremals for the Trudinger-Moser inequalities on the whole space rely on finding a radially maximizing sequence through the symmetry and rearrangement technique. Obviously, this method is not efficient to deal with the existence of maximizers for the double weighted Trudinger-Moser inequality Eq. 1.4 because of the presence of the weight t and ss. In order to overcome this difficulty, we first apply the method of change of variables developed by Dong and Lu (Calc. Var. Part. Diff. Eq. 55, 26-88, 2016) to eliminate the weight ss. Then we can employ the method combining the rearrangement and blow-up analysis to obtain the existence of the extremals to the double weighted Trudinger-Moser inequality Eq. 1.4. By constructing a proper test function sequence, we also derive the sharpness of the exponent a of the Trudinger-Moser inequalities Eqs. 1.3 and 1.4 (see Theorem 1.2 and 1.4). This complements earlier results in Nguyen (2017); Li and Yang (J. Diff. Eq. 264, 4901-4943, 2018); Lu and Zhu (J. Diff. Eq. 267, 3046-3082, 2019).