Sharp Weighted Trudinger-Moser Inequalities with the L ⁿ 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 Lⁿ 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).