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September  2020, 19(9): 4257-4268. doi: 10.3934/cpaa.2020191

On nonexistence of extremals for the Trudinger-Moser functionals involving $ L^p $ norms

Yamin Wang

School of Mathematics, Renmin University of China, Beijing 100872, China

Received  August 2019 Revised  March 2020 Published  June 2020

Let
$ \Omega $
 be the smooth bounded domian in
$ \mathbb{R}^2 $
,
$ W_0^{1, 2}(\Omega) $
 be the standard Sobolev space. We concern a Trudinger-Moser inequality involving
$ L^p $
 norms. For any
$ p>1 $
, denote
$ \lambda_p(\Omega) = \inf\limits_{u\in W_0^{1, 2}(\Omega), u\not\equiv0} \frac{\|\nabla u\|_2^2}{\|u\|_p^2}. $
We prove that for any
$ p>1 $
 and any
$ 0\leq\tau<\lambda_p $
, there exists a positive real number
$ \tau^\ast $
 such that if
$ \tau^\ast <\tau<\lambda_p $
, the supremum
$ \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*} $
can not be achieved by any
$ u\in W_0^{1, 2}(\Omega) $
 with
$ \| \nabla u\|_{2}^2\leq4 \pi $
. This is based on a method of energy estimate, which is developed by [14, 15, 16].
Keywords: Trudinger-Moser inequality, energy estimate, extremal function.
Mathematics Subject Classification: Primary: 35A01, 35B33; Secondary: 35B44.
Citation: Yamin Wang. On nonexistence of extremals for the Trudinger-Moser functionals involving $ L^p $ norms. Communications on Pure and Applied Analysis, 2020, 19 (9) : 4257-4268. doi: 10.3934/cpaa.2020191
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.

[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.

[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.

[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.

[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.

[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.

[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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