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 Poincaré 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.
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Carleson, L., Chang, S.Y. A.: On the existence of an extremal function for an inequality of J. Moser. Bull. Sci. Math. (2) 110(2), 113-127 (1986)
Cassani, D., Sani, F., Tarsi, C.: Equivalent Moser type inequalities in \(R^2\) and the zero mass case. J. Funct. Anal. 267(11), 4236–4263 (2014)
Dong, M., Lam, N., Lu, G.: Sharp weighted Trudinger–Moser inequalities, Caffarelli–Kohn–Nirenberg inequalities and their extremal functions (Preprint)
do Ó, J. M.: N-Laplacian equations in \(R^N\) with critical growth. Abstr. Appl. Anal. 2(3-4), 301-315 (1997)
Ishiwata, M., Nakamura, M., Wadade, H.: On the sharp constant for the weighted Trudinger-Moser type inequality of the scaling invariant form. Ann. Inst. H. Poincaré Anal. Non Linaire 31(2), 297–314 (2014)
Adachi, S., Tanaka, K.: A scale-invariant form of Trudinger-Moser inequality and its best exponent. Proc. Amer. Math. Soc. 1102, 148–153 (1999)
Lam, N., Lu, G.: A new approach to sharp Moser–Trudinger and Adams type inequalities: a rearrangement-free argument. J. Differ. Equa. 255(3), 298–325 (2013)
Lam, N., Lu, G.: Sharp constants and optimizers for a class of the Caffarelli–Kohn–Nirenberg inequalities. arXiv:1510.01224 (Preprint)
Lam, N., Lu, G., Zhang, L.: Equivalence of critical and subcritical sharp Moser–Trudinger–Adams inequalities. arXiv:1504.04858 (Preprint)
Lam, N., Lu, G., Zhang, L.: Existence and nonexistence of extremal functions for sharp Trudinger-Moser inequalities (Preprint)
Li, Y.X., Ruf, B.: A sharp Moser-Trudinger type inequality for unbounded domains in \( \mathbb{R} ^{n}\). Indiana Univ. Math. J. 57(1), 451–480 (2008)
Moser, J.: A sharp form of an inequality by N. Trudinger. Indiana Univ. Math. J. 20, 1077–1092 (1979)
Caffarelli, L., Kohn, R., Nirenberg, L.: First order interpolation inequalities with weights. Compos. Math. 53, 259–275 (1984)
Yudovic, V.I.: Some estimates connected with integral operators and with solutions of elliptic equations. Dokl. Akad. Nauk SSSR 138, 805–808 (1961). (in Russian)
Pohozaev, S.I.: On the eigenfunctions of the equation \(\delta u+\lambda f(u)=0\). Dokl. Akad. Nauk SSSR 165, 36–39 (1965). (in Russian)
Ruf, B.: A sharp Trudinger-Moser type inequality for unbounded domains in \( \mathbb{R}^{2}\). J. Funct. Anal. 219(2), 340–367 (2005)
Trudinger, N.S.: On imbeddings into Orlicz spaces and some applications. J. Math. Mech. 17, 473–483 (1967)
Acknowledgments
The results of this work have been presented by the first author at an invited talk at the AMS special session on Geometric Inequalities and Nonlinear Partial Differential Equations in Las Vegas in April, 2015. Part of the work was done while both authors were visiting Beijing Normal University, China.
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Communicated by N. Trudinger.
Research of this work was partly supported by a US NSF grant DMS-1301595 and a Simons Fellowship from the Simons Foundation.
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Dong, M., Lu, G. Best constants and existence of maximizers for weighted Trudinger–Moser inequalities. Calc. Var. 55, 88 (2016). https://doi.org/10.1007/s00526-016-1014-7
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DOI: https://doi.org/10.1007/s00526-016-1014-7