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A convergent Lagrangian discretization for $ p $-Wasserstein and flux-limited diffusion equations
On nonexistence of extremals for the Trudinger-Moser functionals involving $ L^p $ norms
School of Mathematics, Renmin University of China, Beijing 100872, China |
$ \Omega $ |
$ \mathbb{R}^2 $ |
$ W_0^{1, 2}(\Omega) $ |
$ L^p $ |
$ p>1 $ |
$ \lambda_p(\Omega) = \inf\limits_{u\in W_0^{1, 2}(\Omega), u\not\equiv0} \frac{\|\nabla u\|_2^2}{\|u\|_p^2}. $ |
$ p>1 $ |
$ 0\leq\tau<\lambda_p $ |
$ \tau^\ast $ |
$ \tau^\ast <\tau<\lambda_p $ |
$ \begin{equation*} \sup\limits_{u\in W_0^{1, 2}(\Omega), \, \| \nabla u\|_{2}^2\leq4 \pi}\int_{\Omega}e^{ u^2 (1+\tau\|u\|_p^2)}dx, \end{equation*} $ |
$ u\in W_0^{1, 2}(\Omega) $ |
$ \| \nabla u\|_{2}^2\leq4 \pi $ |
References:
[1] |
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. |
[2] |
Adimurthi and M. Struwe,
Global compactness properties of semilinear elliptic equation with critical exponential growth, J. Funct. Anal., 175 (2000), 125-167.
doi: 10.1006/jfan.2000.3602. |
[3] |
L. Carleson and A. Chang,
On the existence of an extremal function for an inequality of J. Moser, Bull. Sci. Math., 110 (1986), 113-127.
|
[4] |
W. Y. Ding, J. Jost, J. Y. Li and G. F. Wang,
The differential equation $-\Delta u = 8\pi-8\pi h e^u$ on a compact Riemann Surface, Asian J. Math., 1 (1997), 230-248.
doi: 10.4310/AJM.1997.v1.n2.a3. |
[5] |
O. Druet,
Multibumps analysis in dimension 2, quantification of blow-up levels, Duke Math. J., 132 (2006), 217-269.
doi: 10.1215/S0012-7094-06-13222-2. |
[6] |
O. Druet and P. Thizy, Multi-bumps analysis for Trudinger-Moser nonlinearities i-quantification and location of concerntration points, preprint, arXiv: 1710.08811. Google Scholar |
[7] |
M. Flucher,
Extremal functions for Trudinger-Moser inequality in $2$ dimensions, Comment. Math. Helv., 67 (1992), 471-497.
doi: 10.1007/BF02566514. |
[8] |
B. Gidas, W. M. Ni and L. Nirenberg,
Symmetry and related properties via the maximum principle, Commun. Math. Phys., 68 (1979), 209-243.
|
[9] |
Y. X. Li,
Moser-Trudinger inequality on compact Riemannian manifolds of dimension two, J. Partial Differ. Equ., 14 (2001), 163-192.
|
[10] |
Y. X. Li,
The existence of the extremal function of Moser-Trudinger inequality on compact Riemannian manifolds, Sci. China Ser. A, 48 (2005), 618-648.
doi: 10.1360/04ys0050. |
[11] |
K. Lin,
Extremal functions for Moser's inequality, Trans. Amer. Math. Soc., 348 (1996), 2663-2671.
doi: 10.1090/S0002-9947-96-01541-3. |
[12] |
P. L. Lions,
The concentration-compactness principle in the calculus of variation, the limit case, part I, Rev. Mat. Iberoam., 1 (1985), 145-201.
doi: 10.4171/RMI/6. |
[13] |
G. Z. Lu and Y. Y. Yang,
The sharp constant and extremal functions for Moser-Trudinger inequalities involving $L^{p}$ norms, Discrete Contin. Dyn. Syst., 25 (2009), 963-979.
doi: 10.3934/dcds.2009.25.963. |
[14] |
A. Malchiodi and L. Martinazzi,
Critical points of the Moser-Trudinger functional on a disk, J. Eur. Math. Soc., 16 (2014), 893-908.
doi: 10.4171/JEMS/450. |
[15] |
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), 94.
doi: 10.1007/s00526-017-1184-y. |
[16] |
G. Mancini and P. Thizy,
Non-existence of extremals for the Adimurthi-Druet inequality, J. Differ. Equ., 266 (2019), 1051-1072.
doi: 10.1016/j.jde.2018.07.065. |
[17] |
J. Moser,
A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J., 20 (1971), 1077-1091.
doi: 10.1512/iumj.1971.20.20101. |
[18] |
V. H. Nguyen,
Improved Moser-Trudinger inequality of Tintarev type in dimension $n$ and the existence of its extremal functions, Ann. Glob. Anal. Geom., 54 (2018), 237-256.
doi: 10.1007/s10455-018-9599-z. |
[19] |
J. Peetre,
Espaces d'interpolation et thereme de Soboleff, Ann. Inst. Fourier, 16 (1996), 279-317.
|
[20] |
S. Pohozaev, The Sobolev embedding in the special case $pl = n$, in Proceedings of the Technical Scientific Conference on Advances of Scientific Reseach, Mathematics Sections, Moscow, (1965), 158–170. Google Scholar |
[21] |
M. Struwe,
Critical points of embedding of $H_0^1$ into Orlic spaces, Ann. Inst. Henri Poincare Anal. Non Lineaire, 5 (1988), 425-464.
|
[22] |
C. Tintarev,
Trudinger-Moser inequality with remainder terms, J. Funct. Anal., 266 (2014), 55-66.
doi: 10.1016/j.jfa.2013.09.009. |
[23] |
N. Trudinger,
On embeddings into Orlicz space and some applications, J. Math. Mech., 17 (1967), 473-483.
doi: 10.1512/iumj.1968.17.17028. |
[24] |
Y. Y. Yang,
A sharp form of Moser-Trudinger inequality in high dimension, J. Funct. Anal., 239 (2006), 100-126.
doi: 10.1016/j.jfa.2006.06.002. |
[25] |
Y. Y. Yang,
Corrigendum to: "A sharp form of Moser-Trudinger inequality in high dimension", J. Funct. Anal., 242 (2007), 669-671.
doi: 10.1016/j.jfa.2006.09.006. |
[26] |
Y. 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. |
[27] |
Y. Y. Yang,
Extremal functions for Trudinger-Moser inequalities of Adimurthi-Druet type in dimension two, J. Differ. Equ., 258 (2015), 3161-3193.
doi: 10.1016/j.jde.2015.01.004. |
[28] |
Y. Y. Yang, Nonexistence of extremals for an inequality of Adimurthi-Druet on a closed Riemann surface, Sci. China Math., 63 (2020), preprint, arXiv: 1812.05884. Google Scholar |
[29] |
Y. Y. Yang and X. B. Zhu,
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. |
[30] |
Y. Y. Yang,
A remark on energy estimates concerning extremals for Trudinger-Moser inequalities on a disc, Arch. Math. (Basel), 111 (2018), 215-223.
doi: 10.1007/s00013-018-1181-1. |
[31] |
Y. Y. Yang,
Existence of extremals for critical Trudinger-Moser inequalities via the method of energy estimate, J. Math. Anal. Appl., 479 (2019), 1281-1291.
doi: 10.1016/j.jmaa.2019.06.079. |
[32] |
V. I. Yudovich,
Some estimates connected with integral operators and with solutions of elliptic equations, Sov. Math. Docl., 2 (1961), 746-749.
|
[33] |
A. F. Yuan and X. B. Zhu,
An improved singular Trudinger-Moser inequality in unit ball, J. Math. Anal. Appl., 435 (2016), 244-252.
doi: 10.1016/j.jmaa.2015.10.038. |
[34] |
J. Y. Zhu,
Improved Moser-Trudinger Inequality Involving $L^p$ Norm in $n$ Dimensions, Adv. Nonlinear Stud., 14 (2014), 273-293.
doi: 10.1515/ans-2014-0202. |
show all references
References:
[1] |
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. |
[2] |
Adimurthi and M. Struwe,
Global compactness properties of semilinear elliptic equation with critical exponential growth, J. Funct. Anal., 175 (2000), 125-167.
doi: 10.1006/jfan.2000.3602. |
[3] |
L. Carleson and A. Chang,
On the existence of an extremal function for an inequality of J. Moser, Bull. Sci. Math., 110 (1986), 113-127.
|
[4] |
W. Y. Ding, J. Jost, J. Y. Li and G. F. Wang,
The differential equation $-\Delta u = 8\pi-8\pi h e^u$ on a compact Riemann Surface, Asian J. Math., 1 (1997), 230-248.
doi: 10.4310/AJM.1997.v1.n2.a3. |
[5] |
O. Druet,
Multibumps analysis in dimension 2, quantification of blow-up levels, Duke Math. J., 132 (2006), 217-269.
doi: 10.1215/S0012-7094-06-13222-2. |
[6] |
O. Druet and P. Thizy, Multi-bumps analysis for Trudinger-Moser nonlinearities i-quantification and location of concerntration points, preprint, arXiv: 1710.08811. Google Scholar |
[7] |
M. Flucher,
Extremal functions for Trudinger-Moser inequality in $2$ dimensions, Comment. Math. Helv., 67 (1992), 471-497.
doi: 10.1007/BF02566514. |
[8] |
B. Gidas, W. M. Ni and L. Nirenberg,
Symmetry and related properties via the maximum principle, Commun. Math. Phys., 68 (1979), 209-243.
|
[9] |
Y. X. Li,
Moser-Trudinger inequality on compact Riemannian manifolds of dimension two, J. Partial Differ. Equ., 14 (2001), 163-192.
|
[10] |
Y. X. Li,
The existence of the extremal function of Moser-Trudinger inequality on compact Riemannian manifolds, Sci. China Ser. A, 48 (2005), 618-648.
doi: 10.1360/04ys0050. |
[11] |
K. Lin,
Extremal functions for Moser's inequality, Trans. Amer. Math. Soc., 348 (1996), 2663-2671.
doi: 10.1090/S0002-9947-96-01541-3. |
[12] |
P. L. Lions,
The concentration-compactness principle in the calculus of variation, the limit case, part I, Rev. Mat. Iberoam., 1 (1985), 145-201.
doi: 10.4171/RMI/6. |
[13] |
G. Z. Lu and Y. Y. Yang,
The sharp constant and extremal functions for Moser-Trudinger inequalities involving $L^{p}$ norms, Discrete Contin. Dyn. Syst., 25 (2009), 963-979.
doi: 10.3934/dcds.2009.25.963. |
[14] |
A. Malchiodi and L. Martinazzi,
Critical points of the Moser-Trudinger functional on a disk, J. Eur. Math. Soc., 16 (2014), 893-908.
doi: 10.4171/JEMS/450. |
[15] |
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), 94.
doi: 10.1007/s00526-017-1184-y. |
[16] |
G. Mancini and P. Thizy,
Non-existence of extremals for the Adimurthi-Druet inequality, J. Differ. Equ., 266 (2019), 1051-1072.
doi: 10.1016/j.jde.2018.07.065. |
[17] |
J. Moser,
A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J., 20 (1971), 1077-1091.
doi: 10.1512/iumj.1971.20.20101. |
[18] |
V. H. Nguyen,
Improved Moser-Trudinger inequality of Tintarev type in dimension $n$ and the existence of its extremal functions, Ann. Glob. Anal. Geom., 54 (2018), 237-256.
doi: 10.1007/s10455-018-9599-z. |
[19] |
J. Peetre,
Espaces d'interpolation et thereme de Soboleff, Ann. Inst. Fourier, 16 (1996), 279-317.
|
[20] |
S. Pohozaev, The Sobolev embedding in the special case $pl = n$, in Proceedings of the Technical Scientific Conference on Advances of Scientific Reseach, Mathematics Sections, Moscow, (1965), 158–170. Google Scholar |
[21] |
M. Struwe,
Critical points of embedding of $H_0^1$ into Orlic spaces, Ann. Inst. Henri Poincare Anal. Non Lineaire, 5 (1988), 425-464.
|
[22] |
C. Tintarev,
Trudinger-Moser inequality with remainder terms, J. Funct. Anal., 266 (2014), 55-66.
doi: 10.1016/j.jfa.2013.09.009. |
[23] |
N. Trudinger,
On embeddings into Orlicz space and some applications, J. Math. Mech., 17 (1967), 473-483.
doi: 10.1512/iumj.1968.17.17028. |
[24] |
Y. Y. Yang,
A sharp form of Moser-Trudinger inequality in high dimension, J. Funct. Anal., 239 (2006), 100-126.
doi: 10.1016/j.jfa.2006.06.002. |
[25] |
Y. Y. Yang,
Corrigendum to: "A sharp form of Moser-Trudinger inequality in high dimension", J. Funct. Anal., 242 (2007), 669-671.
doi: 10.1016/j.jfa.2006.09.006. |
[26] |
Y. 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. |
[27] |
Y. Y. Yang,
Extremal functions for Trudinger-Moser inequalities of Adimurthi-Druet type in dimension two, J. Differ. Equ., 258 (2015), 3161-3193.
doi: 10.1016/j.jde.2015.01.004. |
[28] |
Y. Y. Yang, Nonexistence of extremals for an inequality of Adimurthi-Druet on a closed Riemann surface, Sci. China Math., 63 (2020), preprint, arXiv: 1812.05884. Google Scholar |
[29] |
Y. Y. Yang and X. B. Zhu,
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. |
[30] |
Y. Y. Yang,
A remark on energy estimates concerning extremals for Trudinger-Moser inequalities on a disc, Arch. Math. (Basel), 111 (2018), 215-223.
doi: 10.1007/s00013-018-1181-1. |
[31] |
Y. Y. Yang,
Existence of extremals for critical Trudinger-Moser inequalities via the method of energy estimate, J. Math. Anal. Appl., 479 (2019), 1281-1291.
doi: 10.1016/j.jmaa.2019.06.079. |
[32] |
V. I. Yudovich,
Some estimates connected with integral operators and with solutions of elliptic equations, Sov. Math. Docl., 2 (1961), 746-749.
|
[33] |
A. F. Yuan and X. B. Zhu,
An improved singular Trudinger-Moser inequality in unit ball, J. Math. Anal. Appl., 435 (2016), 244-252.
doi: 10.1016/j.jmaa.2015.10.038. |
[34] |
J. Y. Zhu,
Improved Moser-Trudinger Inequality Involving $L^p$ Norm in $n$ Dimensions, Adv. Nonlinear Stud., 14 (2014), 273-293.
doi: 10.1515/ans-2014-0202. |
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