Remarks on the Moser–Trudinger type inequality with logarithmic weights in dimension 𝑁

Nguyen, Van

doi:10.1090/proc/14566

Remarks on the Moser–Trudinger type inequality with logarithmic weights in dimension $N$
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Proc. Amer. Math. Soc. 147 (2019), 5183-5193 Request permission

Abstract:

We provide a simpler proof of the Moser–Trudinger type inequality with logarithmic weight $w_\beta = (-\ln |x|)^{\beta (N-1)}$, $\beta \in [0,1)$ in dimension $N\geq 2$ recently established by Calanchi and Ruf. Our proof is based on a suitable change of functions on $B$ and the classical Moser–Trudinger inequality on $B$. We also prove the existence of maximizers for this inequality when $\beta \geq 0$ sufficiently small.

References

Adimurthi and K. Sandeep, A singular Moser-Trudinger embedding and its applications, NoDEA Nonlinear Differential Equations Appl. 13 (2007), no. 5-6, 585–603. MR 2329019, DOI 10.1007/s00030-006-4025-9
Adimurthi and Cyril Tintarev, On compactness in the Trudinger-Moser inequality, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 13 (2014), no. 2, 399–416. MR 3235520
Marta Calanchi, Some weighted inequalities of Trudinger-Moser type, Analysis and topology in nonlinear differential equations, Progr. Nonlinear Differential Equations Appl., vol. 85, Birkhäuser/Springer, Cham, 2014, pp. 163–174. MR 3330728
Marta Calanchi and Bernhard Ruf, On Trudinger-Moser type inequalities with logarithmic weights, J. Differential Equations 258 (2015), no. 6, 1967–1989. MR 3302527, DOI 10.1016/j.jde.2014.11.019
Marta Calanchi and Bernhard Ruf, Trudinger-Moser type inequalities with logarithmic weights in dimension $N$, Nonlinear Anal. 121 (2015), 403–411. MR 3348931, DOI 10.1016/j.na.2015.02.001
Lennart Carleson and Sun-Yung A. Chang, On the existence of an extremal function for an inequality of J. Moser, Bull. Sci. Math. (2) 110 (1986), no. 2, 113–127 (English, with French summary). MR 878016
Gyula Csató and Prosenjit Roy, Extremal functions for the singular Moser-Trudinger inequality in 2 dimensions, Calc. Var. Partial Differential Equations 54 (2015), no. 2, 2341–2366. MR 3396455, DOI 10.1007/s00526-015-0867-5
Gyula Csató and Prosenjit Roy, Singular Moser-Trudinger inequality on simply connected domains, Comm. Partial Differential Equations 41 (2016), no. 5, 838–847. MR 3508324, DOI 10.1080/03605302.2015.1123276
G. Csató, P. Roy, and V. H. Nguyen, Extremals for the singular Moser–Trudinger inequality via $n$-harmonic transplantation\text, preprint, arXiv:1801.03932v3, 2018.
Djairo G. de Figueiredo, João Marcos do Ó, and Bernhard Ruf, On an inequality by N. Trudinger and J. Moser and related elliptic equations, Comm. Pure Appl. Math. 55 (2002), no. 2, 135–152. MR 1865413, DOI 10.1002/cpa.10015
Martin Flucher, Extremal functions for the Trudinger-Moser inequality in $2$ dimensions, Comment. Math. Helv. 67 (1992), no. 3, 471–497. MR 1171306, DOI 10.1007/BF02566514
Nguyen Lam and Guozhen Lu, A new approach to sharp Moser-Trudinger and Adams type inequalities: a rearrangement-free argument, J. Differential Equations 255 (2013), no. 3, 298–325. MR 3053467, DOI 10.1016/j.jde.2013.04.005
M. A. Leckband, An integral inequality with applications, Trans. Amer. Math. Soc. 283 (1984), no. 1, 157–168. MR 735413, DOI 10.1090/S0002-9947-1984-0735413-7
Yuxiang Li and Bernhard Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in $\Bbb R^n$, Indiana Univ. Math. J. 57 (2008), no. 1, 451–480. MR 2400264, DOI 10.1512/iumj.2008.57.3137
Kai-Ching Lin, Extremal functions for Moser’s inequality, Trans. Amer. Math. Soc. 348 (1996), no. 7, 2663–2671. MR 1333394, DOI 10.1090/S0002-9947-96-01541-3
P.-L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. II, Rev. Mat. Iberoamericana 1 (1985), no. 2, 45–121. MR 850686, DOI 10.4171/RMI/12
Guozhen Lu and Hanli Tang, Best constants for Moser-Trudinger inequalities on high dimensional hyperbolic spaces, Adv. Nonlinear Stud. 13 (2013), no. 4, 1035–1052. MR 3115151, DOI 10.1515/ans-2013-0415
Andrea Malchiodi and Luca Martinazzi, Critical points of the Moser-Trudinger functional on a disk, J. Eur. Math. Soc. (JEMS) 16 (2014), no. 5, 893–908. MR 3210956, DOI 10.4171/JEMS/450
G. Mancini and K. Sandeep, Moser-Trudinger inequality on conformal discs, Commun. Contemp. Math. 12 (2010), no. 6, 1055–1068. MR 2748285, DOI 10.1142/S0219199710004111
Gianni Mancini, Kunnath Sandeep, and Cyril Tintarev, Trudinger-Moser inequality in the hyperbolic space ${\Bbb H}^N$, Adv. Nonlinear Anal. 2 (2013), no. 3, 309–324. MR 3089744, DOI 10.1515/anona-2013-0001
J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J. 20 (1970/71), 1077–1092. MR 301504, DOI 10.1512/iumj.1971.20.20101
S. I. Pohožaev, On the eigenfunctions of the equation $\Delta u+\lambda f(u)=0$, Dokl. Akad. Nauk SSSR 165 (1965), 36–39 (Russian). MR 0192184
Prosenjit Roy, Extremal function for Moser-Trudinger type inequality with logarithmic weight, Nonlinear Anal. 135 (2016), 194–204. MR 3473116, DOI 10.1016/j.na.2016.01.024
Bernhard Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in $\Bbb R^2$, J. Funct. Anal. 219 (2005), no. 2, 340–367. MR 2109256, DOI 10.1016/j.jfa.2004.06.013
Michael Struwe, Critical points of embeddings of $H^{1,n}_0$ into Orlicz spaces, Ann. Inst. H. Poincaré Anal. Non Linéaire 5 (1988), no. 5, 425–464 (English, with French summary). MR 970849, DOI 10.1016/S0294-1449(16)30338-9
Neil S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech. 17 (1967), 473–483. MR 0216286, DOI 10.1512/iumj.1968.17.17028
Guofang Wang and Dong Ye, A Hardy-Moser-Trudinger inequality, Adv. Math. 230 (2012), no. 1, 294–320. MR 2900545, DOI 10.1016/j.aim.2011.12.001
V. I. Judovič, Some estimates connected with integral operators and with solutions of elliptic equations, Dokl. Akad. Nauk SSSR 138 (1961), 805–808 (Russian). MR 0140822