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On attainability of Moser-Trudinger inequality with logarithmic weights in higher dimensions
Department of Mathematics and Statistics, Indian Institute of Technology, Kanpur, 208016, India |
$ \beta \in [0,1) $ |
$ w_0(x) = |\log |x||^{\beta(n-1)} $ |
$ \left( \log \frac{e}{|x|}\right)^{\beta(n-1)} $ |
$ \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 $ |
$ \alpha \leq \alpha_\beta = n\left[\omega_{n-1}^{\frac{1}{n-1}}(1-\beta) \right]^{\frac{1}{1-\beta}} $ |
$ \omega_{n-1} $ |
$ \mathbb {R}^n $ |
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 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. |
[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. 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. |
[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 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. |
[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. 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. |
[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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