The Hardy–Moser–Trudinger inequality via the transplantation of Green functions

Van Hoang Nguyen

doi:10.3934/cpaa.2020155

Article Contents

2020, Volume 19, Issue 7: 3559-3574. Doi: 10.3934/cpaa.2020155

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The Hardy–Moser–Trudinger inequality via the transplantation of Green functions

Van Hoang Nguyen

Department of Mathematics, FPT University, Ha Noi, Vietnam

Received: March 31, 2019

Revised: December 31, 2019

Published: April 2020

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Abstract

We provide a new proof of the Hardy–Moser–Trudinger inequality and the existence of its extremals which are established by Wang and Ye ("G. Wang, and D. Ye, A Hardy–Moser–Trudinger inequality, Adv. Math, 230 (2012) 294–230.") without using the blow-up analysis method. Our proof is based on the transformation of functions via the transplantation of Green functions. This method enables us to compute explicitly the concentrating level of the Hardy–Moser–Trudinger functional over the normalizing concentrating sequences which is crucial to prove the existence of extremals for the Hardy–Moser–Trudinger inequality. Some comments on the applications of this approach to some other Moser–Trudinger type inequalities are given.

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Mathematics Subject Classification: Primary: 26D10; Secondary: 35A23, 46E35.

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[1]	S. Adachi and K. Tanaka, Trudinger type inequalities in ${\mathbb R}^N$ and their best exponents, Proc. Amer. Math. Soc., 128 (2000), 2051-2057. doi: 10.1090/S0002-9939-99-05180-1.
[2]	D. R. Adams, A sharp inequality of J. Moser for higher order derivatives, Ann. Math., 128 (1988), 385-398. doi: 10.2307/1971445.
[3]	Adimurthi and O. Druet, Blow-up analysis in dimension $2$ and a sharp form of Trudinger-Moser inequality, Commun. Partial Differ. Equ., 29 (2004), 295-322. doi: 10.1081/PDE-120028854.
[4]	Ad imurthi and K. Sandeep, A singular Moser–Trudinger embedding and its applications, NoDea Nonlinear Differ. Equ. Appl., 13 (2007), 585-603. doi: 10.1007/s00030-006-4025-9.
[5]	Adimurthi and C. Tintarev, On a version of Trudinger-Moser inequality with Möbius shift invariance, Calc. Var. Partial Differ. Equ., 39 (2010), 203-212. doi: 10.1007/s00526-010-0307-5.
[6]	Adimurthi and Y. Yang, An interpolation of Hardy inequality and Trundinger–Moser inequality in ${\mathbb R}^N$ and its applications, Int. Math. Res. Not. IMRN, (2010), 2394–2426. doi: 10.1093/imrn/rnp194.
[7]	R. D. Benguria, R. L. Frank and M. Loss, The sharp constant in the Hardy-Sobolev-Maz'ya inequality in the three dimensional upper half-space, Math. Res. Lett., 15 (2008), 613-622. doi: 10.4310/MRL.2008.v15.n4.a1.
[8]	L. Carleson and S. Y. A. Chang, On the existence of an extremal function for an inequality of J. Moser, Bull. Sci. Math., 110 (1986), 113-127.
[9]	G. Csató and P. Roy, Extremal functions for the singular Moser-Trudinger inequality in $2$ dimensions, Calc. Var. Partial Differ. Equ., 54 (2015), 2341-2366. doi: 10.1007/s00526-015-0867-5.
[10]	G. Csató and P. Roy, Singular Moser-Trudinger inequality on simply connected domains, Commun. Partial Differ. Equ., 41 (2016), 838-847. doi: 10.1080/03605302.2015.1123276.
[11]	G. Csató, V. H. Nguyen and P. Roy, Extremals for the singular Moser-Trudinger inequality via $n$-harmonic transplantation, preprint, arXiv: 1801.03932v3.
[12]	D. G. De Figueiredo, J. M. do Ó and B. Ruf, On an inequality by N. Trudinger and J. Moser and related elliptic equations, Commun. Pure Appl. Math., 55 (2002), 135-152. doi: 10.1002/cpa.10015.
[13]	M. Flucher, Extremal functions for the Trudinger-Moser inequality in $2$ dimensions, Comment. Math. Helv., 67 (1992), 471-497. doi: 10.1007/BF02566514.
[14]	M. Ishiwata, Existence and nonexistence of maximizers for variational problems associated with Trudinger–Moser inequalities in $\mathbb R^N$, Math. Ann., 351 (2011), 781-804. doi: 10.1007/s00208-010-0618-z.
[15]	N. Lam and G. Lu, A new approach to sharp Moser-Trudinger and Adams type inequalities: A rearrangement–free argument, J. Differ. Equ., 255 (2013), 298-325. doi: 10.1016/j.jde.2013.04.005.
[16]	Y. Li, Moser-Trudinger inequaity on compact Riemannian manifolds of dimension two, J. Partial Differ. Equ., 14 (2001), 163-192.
[17]	Y. Li, Extremal functions for the Moser-Trudinger inequalities on compact Riemannian manifolds, Sci. China Ser. A, 48 (2005), 618-648. doi: 10.1360/04ys0050.
[18]	Y. Li and B. Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in ${\mathbb R}^n$, Indiana Univ. Math. J., 57 (2008), 451-480. doi: 10.1512/iumj.2008.57.3137.
[19]	J. Li, G. Lu and Q. Yang, Fourier analysis and optimal Hardy-Adams inequalities on hyperbolic spaces of any even dimension, Adv. Math., 333 (2018), 350-385. doi: 10.1016/j.aim.2018.05.035.
[20]	K. Lin, Extremal functions for Moser's inequality, Trans. Amer. Math. Soc., 348 (1996), 2663-2671. doi: 10.1090/S0002-9947-96-01541-3.
[21]	P. L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. II, Rev. Mat. Iberoam., 1 (1985), 45-121. doi: 10.4171/RMI/12.
[22]	G. Lu and Q. Yang, A sharp Trudinger-Moser inequality on any bounded and convex planar domain, Calc. Var. Partial Differ. Equ., 55 (2016), Art. 153, 16 pp. doi: 10.1007/s00526-016-1077-5.
[23]	G. Lu and Q. Yang, Sharp Hardy–Adams inequalities for bi-Laplacian on hyperbolic space of dimension four, Adv. Math., 319 (2017), 567-598. doi: 10.1016/j.aim.2017.08.014.
[24]	G. Mancini and K. Sandeep, Moser-Trudinger inequality on conformal discs, Commun. Contemp. Math., 12 (2010), 1055-1068. doi: 10.1142/S0219199710004111.
[25]	G. Mancini, K. Sandeep and C. Tintarev, Trudinger-Moser inequality in the hyperbolic space $\mathbb H^n$, Adv. Nonlinear Anal., 2 (2013), 309-324. doi: 10.1515/anona-2013-0001.
[26]	G. Mancini and L. Martinazzi, The Moser-Trudinger inequality and its extremals on a disk via energy estimates, Calc. Var. Partial Differ. Equ., 56 (2017), Art. 94, 26 pp. doi: 10.1007/s00526-017-1184-y.
[27]	L. Martinazzi, Fractional Adams-Moser-Trudinger type inequalities, Nonlinear Anal., 127 (2015), 263-278. doi: 10.1016/j.na.2015.06.034.
[28]	V. G. Maz'ya, Sobolev spaces, Springer Verlag, Berlin, New York, 1985. doi: 10.1007/978-3-662-09922-3.
[29]	J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J., 20 (1970/71), 1077-1092. doi: 10.1512/iumj.1971.20.20101.
[30]	V. H. Nguyen, The sharp Poincaré–Sobolev type inequalities in the hyperbolic spaces $\mathbb H^n$, J. Math. Anal. Appl., 462 (2018), 1570-1584. doi: 10.1016/j.jmaa.2018.02.054.
[31]	V. H. Nguyen, Improved Moser-Trudinger type inequalities in the hyperbolic space $\mathbb H^n$, Nonlinear Anal., 168 (2018), 67-80. doi: 10.1016/j.na.2017.11.009.
[32]	V. H. Nguyen, Improved Moser–Trudinger inequality of Tintarev type in dimension $n$ and the existence of its extremal functions, Ann. Global Anal. Geom., 54 (2018), 237-256. doi: 10.1007/s10455-018-9599-z.
[33]	V. H. Nguyen, Improved singular Moser-Trudinger inequalities and their extremal functions, Potential Anal., in press.
[34]	V. H. Nguyen, The sharp Hardy–Moser–Trudinger inequality in dimension $n$, preprint, arXiv: 1909.12587.
[35]	S. I. Pohožaev, On the eigenfunctions of the equation $\Delta u + \lambda f(u) = 0$] (Russian), Dokl. Akad. Nauk. SSSR, 165 (1965), 36-39.
[36]	B. Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in ${\mathbb R}^2$, J. Funct. Anal., 219 (2005), 340-367. doi: 10.1016/j.jfa.2004.06.013.
[37]	A. Tertikas and C. Tintarev, On existence of minimizers for the Hardy-Sobolev-Maz'ya inequality, Ann. Mat. Pura Appl. (4), 186 (2007), 645-662. doi: 10.1007/s10231-006-0024-z.
[38]	C. Tintarev, Trudinger–Moser inequality with remainder terms, J. Funct. Anal., 266 (2014), 55-66. doi: 10.1016/j.jfa.2013.09.009.
[39]	N. S. Trudinger, On imbedding into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-483. doi: 10.1512/iumj.1968.17.17028.
[40]	G. Wang and D. Ye, A Hardy-Moser-Trudinger inequality, Adv. Math., 230 (2012), 294-320. doi: 10.1016/j.aim.2011.12.001.
[41]	X. Wang, Improved Hardy-Adams inequality on hyperbolic space of dimension four, Nonlinear Anal., 182 (2019), 45-56. doi: 10.1016/j.na.2018.12.007.
[42]	X. Wang, Singular Hardy-Moser-Trudinger inequality and the existence of extremals on the unit disc, Commun. Pure Appl. Anal., 18 (2019), 2741-2757. doi: 10.3934/cpaa.2019121.
[43]	Y. Yang, A sharp form of the Moser–Trudinger inequality on a compact Riemannian surface, Trans. Amer. Math. Soc., 359 (2007), 5761-5776. doi: 10.1090/S0002-9947-07-04272-9.
[44]	Y. Yang and X. Zhu, An improved Hardy-Trudinger-Moser inequality, Ann. Global Anal. Geom., 49 (2016), 23-41. doi: 10.1007/s10455-015-9478-9.
[45]	Q. Yang, D. Su and Y. Kong, Sharp Moser-Trudinger inequalities on Riemannian manifolds with negative curvature, Ann. Mat. Pura Appl., 195 (2016), 459-471. doi: 10.1007/s10231-015-0472-4.
[46]	V. I. Yudovič, Some estimates connected with integral operators and with solutions of elliptic equations (Russian), Dokl. Akad. Nauk. SSSR, 138 (1961), 805-808.