September  2019, 39(9): 5207-5222. doi: 10.3934/dcds.2019212

On attainability of Moser-Trudinger inequality with logarithmic weights in higher dimensions

Department of Mathematics and Statistics, Indian Institute of Technology, Kanpur, 208016, India

Received  August 2018 Revised  March 2019 Published  May 2019

Fund Project: Part of this research is supported by Inspire programme under the contract number IFA14/MA-43

Moser-Trudinger inequality was generalised by Calanchi-Ruf to the following version: If
$ \beta \in [0,1) $
and
$ w_0(x) = |\log |x||^{\beta(n-1)} $
or
$ \left( \log \frac{e}{|x|}\right)^{\beta(n-1)} $
then
$ \sup\limits_{\int_B | \nabla u|^nw_0 \leq 1 , u \in W_{0,rad}^{1,n}(w_0,B)} \;\; \int_B \exp\left(\alpha |u|^{\frac{n}{(n-1)(1-\beta)}} \;\;\; \right) dx < \infty $
if and only if
$ \alpha \leq \alpha_\beta = n\left[\omega_{n-1}^{\frac{1}{n-1}}(1-\beta) \right]^{\frac{1}{1-\beta}} $
where
$ \omega_{n-1} $
denotes the surface measure of the unit sphere in
$ \mathbb {R}^n $
. The primary goal of this work is to address the issue of existence of extremal function for the above inequality. A non-existence (of extremal function) type result is also discussed, for the usual Moser-Trudinger functional.
Citation: Prosenjit Roy. On attainability of Moser-Trudinger inequality with logarithmic weights in higher dimensions. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5207-5222. doi: 10.3934/dcds.2019212
References:
[1]

Adimurthi and K. Sandeep, A singular Moser-Trudinger embedding and its applications, NoDEA Nonlinear Differential Equations Appl., 13 (2007), 585-603. doi: 10.1007/s00030-006-4025-9.

[2]

Adimurthi and C. Tintarev, On compactness in the Trudinger-Moser inequality, Ann. Sc. Norm. Super. Pisa Cl. Sci., 13 (2014), 399-416.

[3]

M. Calanchi, Some weighted inequalities of Trudinger–Moser type, Progress in Nonlinear Differential Equations and Appl., 85 (2014), 163-174.

[4]

M. Calanchi and B. Ruf, Trudinger-Moser type inequalities with logarithmic weights in dimension $N$, Nonlinear Anal., 121 (2015), 403-411. doi: 10.1016/j.na.2015.02.001.

[5]

M. Calanchi and B. Ruf, On Trudinger-Moser type inequalities with logarithmic weights, J. Differential Equations, 258 (2015), 1967-1989. doi: 10.1016/j.jde.2014.11.019.

[6]

L. Carleson and S.-Y. A. Chang, On the existence of an extremal function for an inequality by J. Moser, Bull. Sci. Math., 110 (1986), 113-127.

[7]

G. Csató and P. Roy, Extremal functions for the singular Moser-Trudinger inequality in 2 dimensions, Calc. Var. Partial Differential Equations, 54 (2015), 2341-2366. doi: 10.1007/s00526-015-0867-5.

[8]

G. Csató and P. Roy, The singular Moser-Trudinger inequality on simply connected domain, Comm. Partial Differential Equations, 41 (2016), 838-847. doi: 10.1080/03605302.2015.1123276.

[9]

G. Csató, N. H. Nguyen and P. Roy, Extremals for the Singular Moser-Trudinger Inequality via n-Harmonic Transplantation, preprint, arXiv: 1801.03932v3.

[10]

D. G. de FigueiredoJ. M. do O and B. Ruf, Elliptic equations and systems with critical Trudinger-Moser nonlinearities., Discrete Contin. Dyn. Syst., 30 (2011), 455-476. doi: 10.3934/dcds.2011.30.455.

[11]

M. Flucher, Extremal functions for the Trudinger-Moser inequality in 2 dimensions, Comment. Math. Helvetici, 67 (1992), 471-497. doi: 10.1007/BF02566514.

[12]

N. Lam, Equivalence of sharp Trudinger-Moser-Adams inequalities, Commun. Pure Appl. Anal, 16 (2017), 973-997. doi: 10.3934/cpaa.2017047.

[13]

N. Lam and G. Lu, A new approach to sharp Moser-Trudinger and Adams type inequalities: A rearrangement-free argument., J. Differential Equations, 255 (2013), 298-325. doi: 10.1016/j.jde.2013.04.005.

[14]

X. Li and Y. Yang, Extremal functions for singular Trudinger-Moser inequalities in the entire Euclidean space, prerpint, arXiv: 1612.08247

[15]

K.-C. Lin, Extremal functions for Moser's inequality, Trans. Amer. Math. Soc., 348 (1996), 2663-2671. doi: 10.1090/S0002-9947-96-01541-3.

[16]

P.-L. Lions, The concentration-compactness principle in the Calculus of variations. The limit case. Ⅰ, Rev. Mat. Iberoamericana, 1 (1985), 145-201. doi: 10.4171/RMI/6.

[17]

G. Lu and H. Tang, Best constants for Moser-Trudinger inequalities on high dimensional hyperbolic spaces, Adv. Nonlinear Stud., 13 (2013), 1035-1052. doi: 10.1515/ans-2013-0415.

[18]

G. Lu and Y. Yang, Sharp constant and extremal function for the improved Moser-Trudinger inequality involving $L^p$ norm in two dimension, Discrete Contin. Dyn. Syst., 25 (2009), 963-979. doi: 10.3934/dcds.2009.25.963.

[19]

S. Lula and G. Mancini, Extremal functions for singular Moser-Trudinger embeddings, Nonlinear Anal., 156 (2017), 215-248. doi: 10.1016/j.na.2017.02.029.

[20]

A. Malchiodi and L. Martinazzi, Critical points of the Moser-Trudinger functional on a disk, J. Eur. Math. Soc. (JEMS), 16 (2014), 893-908. doi: 10.4171/JEMS/450.

[21]

G. Mancini and L. Battaglia, Remarks on the Moser-Trudinger inequality, Adv. Nonlinear Anal, 2 (2013), 389-425. doi: 10.1515/anona-2013-0014.

[22]

G. Mancini and K. Sandeep, Moser-Trudinger inequality on conformal discs, Commun. Contemp. Math, 12 (2010), 1055-1068. doi: 10.1142/S0219199710004111.

[23]

G. ManciniK. Sandeep and C. Tintarev, Trudinger-Moser inequality in the hyperbolic space $H^N$, Adv. Nonlinear Anal, 2 (2013), 309-324. doi: 10.1515/anona-2013-0001.

[24]

J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J, 20 (1971), 1077-1092. doi: 10.1512/iumj.1971.20.20101.

[25]

Q.-A. Ngo and V. H. Nguyen, An improved Moser-Trudinger inequality involving the first non-zero Neumann eigenvalue with mean value zero in $\mathbb{R}^2$, preprint, arXiv: 1702.08883

[26]

V. H. Nguyen, A sharp Adams inequality in dimension four and its extremal functions, arXiv: 1701.08249

[27]

V. H. Nguyen and F. Takahashi, On a weighted Trudinger-Moser type inequality on the whole space and its (non-)existence of maximizers, Differential Integral Equations, 31 (2018), 785-806.

[28]

P. Roy, Extremal function for Moser-Trudinger type inequality with logarithmic weight, Nonlinear Anal., 135 (2016), 194-204. doi: 10.1016/j.na.2016.01.024.

[29]

B. Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in $R^2$, J. Funct. Anal, 219 (2005), 340-367. doi: 10.1016/j.jfa.2004.06.013.

[30]

M. Struwe, Critical points of embeddings of $H_0^{1, n}$ into Orlicz spaces, Ann. Inst. H. Poincaré Anal. Non Linéaire, 5 (1988), 425-464. doi: 10.1016/S0294-1449(16)30338-9.

[31]

N. S. Trudinger, On embeddings into Orlicz spaces and some applications, J. Math. Mech, 17 (1967), 473-484. doi: 10.1512/iumj.1968.17.17028.

[32]

Y. Yang, Extremal functions for a sharp Moser-Trudinger inequality, Internat. J. Math., 17 (2006), 331-338. doi: 10.1142/S0129167X06003503.

[33]

Y. Yang, Extremal functions for Moser-Trudinger inequalities on 2-dimensional compact Riemannian manifolds with boundary, Internat. J. Math., 17 (2006), 313-330. doi: 10.1142/S0129167X06003473.

[34]

Y. Yang, Extremal functions for Trudinger-Moser inequalities of Adimurthi-Druet type in dimension two, J. Differential Equations, 258 (2015), 3161-3193. doi: 10.1016/j.jde.2015.01.004.

[35]

Y. Yang, A Trudinger-Moser inequality on a compact Riemannian surface involving Gaussian curvature, J. Geom. Anal., 26 (2016), 2893-2913. doi: 10.1007/s12220-015-9653-z.

[36]

X. Zhu and Y. Yang, Blow-up analysis concerning singular Trudinger Moser inequalities in dimension two, J. Funct. Anal, 272 (2017), 3347-3374. doi: 10.1016/j.jfa.2016.12.028.

show all references

References:
[1]

Adimurthi and K. Sandeep, A singular Moser-Trudinger embedding and its applications, NoDEA Nonlinear Differential Equations Appl., 13 (2007), 585-603. doi: 10.1007/s00030-006-4025-9.

[2]

Adimurthi and C. Tintarev, On compactness in the Trudinger-Moser inequality, Ann. Sc. Norm. Super. Pisa Cl. Sci., 13 (2014), 399-416.

[3]

M. Calanchi, Some weighted inequalities of Trudinger–Moser type, Progress in Nonlinear Differential Equations and Appl., 85 (2014), 163-174.

[4]

M. Calanchi and B. Ruf, Trudinger-Moser type inequalities with logarithmic weights in dimension $N$, Nonlinear Anal., 121 (2015), 403-411. doi: 10.1016/j.na.2015.02.001.

[5]

M. Calanchi and B. Ruf, On Trudinger-Moser type inequalities with logarithmic weights, J. Differential Equations, 258 (2015), 1967-1989. doi: 10.1016/j.jde.2014.11.019.

[6]

L. Carleson and S.-Y. A. Chang, On the existence of an extremal function for an inequality by J. Moser, Bull. Sci. Math., 110 (1986), 113-127.

[7]

G. Csató and P. Roy, Extremal functions for the singular Moser-Trudinger inequality in 2 dimensions, Calc. Var. Partial Differential Equations, 54 (2015), 2341-2366. doi: 10.1007/s00526-015-0867-5.

[8]

G. Csató and P. Roy, The singular Moser-Trudinger inequality on simply connected domain, Comm. Partial Differential Equations, 41 (2016), 838-847. doi: 10.1080/03605302.2015.1123276.

[9]

G. Csató, N. H. Nguyen and P. Roy, Extremals for the Singular Moser-Trudinger Inequality via n-Harmonic Transplantation, preprint, arXiv: 1801.03932v3.

[10]

D. G. de FigueiredoJ. M. do O and B. Ruf, Elliptic equations and systems with critical Trudinger-Moser nonlinearities., Discrete Contin. Dyn. Syst., 30 (2011), 455-476. doi: 10.3934/dcds.2011.30.455.

[11]

M. Flucher, Extremal functions for the Trudinger-Moser inequality in 2 dimensions, Comment. Math. Helvetici, 67 (1992), 471-497. doi: 10.1007/BF02566514.

[12]

N. Lam, Equivalence of sharp Trudinger-Moser-Adams inequalities, Commun. Pure Appl. Anal, 16 (2017), 973-997. doi: 10.3934/cpaa.2017047.

[13]

N. Lam and G. Lu, A new approach to sharp Moser-Trudinger and Adams type inequalities: A rearrangement-free argument., J. Differential Equations, 255 (2013), 298-325. doi: 10.1016/j.jde.2013.04.005.

[14]

X. Li and Y. Yang, Extremal functions for singular Trudinger-Moser inequalities in the entire Euclidean space, prerpint, arXiv: 1612.08247

[15]

K.-C. Lin, Extremal functions for Moser's inequality, Trans. Amer. Math. Soc., 348 (1996), 2663-2671. doi: 10.1090/S0002-9947-96-01541-3.

[16]

P.-L. Lions, The concentration-compactness principle in the Calculus of variations. The limit case. Ⅰ, Rev. Mat. Iberoamericana, 1 (1985), 145-201. doi: 10.4171/RMI/6.

[17]

G. Lu and H. Tang, Best constants for Moser-Trudinger inequalities on high dimensional hyperbolic spaces, Adv. Nonlinear Stud., 13 (2013), 1035-1052. doi: 10.1515/ans-2013-0415.

[18]

G. Lu and Y. Yang, Sharp constant and extremal function for the improved Moser-Trudinger inequality involving $L^p$ norm in two dimension, Discrete Contin. Dyn. Syst., 25 (2009), 963-979. doi: 10.3934/dcds.2009.25.963.

[19]

S. Lula and G. Mancini, Extremal functions for singular Moser-Trudinger embeddings, Nonlinear Anal., 156 (2017), 215-248. doi: 10.1016/j.na.2017.02.029.

[20]

A. Malchiodi and L. Martinazzi, Critical points of the Moser-Trudinger functional on a disk, J. Eur. Math. Soc. (JEMS), 16 (2014), 893-908. doi: 10.4171/JEMS/450.

[21]

G. Mancini and L. Battaglia, Remarks on the Moser-Trudinger inequality, Adv. Nonlinear Anal, 2 (2013), 389-425. doi: 10.1515/anona-2013-0014.

[22]

G. Mancini and K. Sandeep, Moser-Trudinger inequality on conformal discs, Commun. Contemp. Math, 12 (2010), 1055-1068. doi: 10.1142/S0219199710004111.

[23]

G. ManciniK. Sandeep and C. Tintarev, Trudinger-Moser inequality in the hyperbolic space $H^N$, Adv. Nonlinear Anal, 2 (2013), 309-324. doi: 10.1515/anona-2013-0001.

[24]

J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J, 20 (1971), 1077-1092. doi: 10.1512/iumj.1971.20.20101.

[25]

Q.-A. Ngo and V. H. Nguyen, An improved Moser-Trudinger inequality involving the first non-zero Neumann eigenvalue with mean value zero in $\mathbb{R}^2$, preprint, arXiv: 1702.08883

[26]

V. H. Nguyen, A sharp Adams inequality in dimension four and its extremal functions, arXiv: 1701.08249

[27]

V. H. Nguyen and F. Takahashi, On a weighted Trudinger-Moser type inequality on the whole space and its (non-)existence of maximizers, Differential Integral Equations, 31 (2018), 785-806.

[28]

P. Roy, Extremal function for Moser-Trudinger type inequality with logarithmic weight, Nonlinear Anal., 135 (2016), 194-204. doi: 10.1016/j.na.2016.01.024.

[29]

B. Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in $R^2$, J. Funct. Anal, 219 (2005), 340-367. doi: 10.1016/j.jfa.2004.06.013.

[30]

M. Struwe, Critical points of embeddings of $H_0^{1, n}$ into Orlicz spaces, Ann. Inst. H. Poincaré Anal. Non Linéaire, 5 (1988), 425-464. doi: 10.1016/S0294-1449(16)30338-9.

[31]

N. S. Trudinger, On embeddings into Orlicz spaces and some applications, J. Math. Mech, 17 (1967), 473-484. doi: 10.1512/iumj.1968.17.17028.

[32]

Y. Yang, Extremal functions for a sharp Moser-Trudinger inequality, Internat. J. Math., 17 (2006), 331-338. doi: 10.1142/S0129167X06003503.

[33]

Y. Yang, Extremal functions for Moser-Trudinger inequalities on 2-dimensional compact Riemannian manifolds with boundary, Internat. J. Math., 17 (2006), 313-330. doi: 10.1142/S0129167X06003473.

[34]

Y. Yang, Extremal functions for Trudinger-Moser inequalities of Adimurthi-Druet type in dimension two, J. Differential Equations, 258 (2015), 3161-3193. doi: 10.1016/j.jde.2015.01.004.

[35]

Y. Yang, A Trudinger-Moser inequality on a compact Riemannian surface involving Gaussian curvature, J. Geom. Anal., 26 (2016), 2893-2913. doi: 10.1007/s12220-015-9653-z.

[36]

X. Zhu and Y. Yang, Blow-up analysis concerning singular Trudinger Moser inequalities in dimension two, J. Funct. Anal, 272 (2017), 3347-3374. doi: 10.1016/j.jfa.2016.12.028.

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