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

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

Prosenjit Roy

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.
Keywords: Moser Trudinger embedding, extremal function.
Mathematics Subject Classification: Primary: 35B38, 35J20, 47N20, 26D10; Secondary: 46E35.
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.  Google Scholar

[2]

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

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[10]

D. G. de Figueiredo, J. 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.  Google Scholar

[11]

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

[12]

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

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[23]

G. Mancini, K. 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.  Google Scholar

[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.  Google Scholar

[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 Google Scholar

[26]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

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

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[10]

D. G. de Figueiredo, J. 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.  Google Scholar

[11]

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

[12]

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

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[23]

G. Mancini, K. 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.  Google Scholar

[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.  Google Scholar

[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 Google Scholar

[26]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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