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

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

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].
Citation: Yamin Wang. On nonexistence of extremals for the Trudinger-Moser functionals involving $ L^p $ norms. Communications on Pure & 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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[4]

W. Y. DingJ. JostJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

B. GidasW. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Commun. Math. Phys., 68 (1979), 209-243.   Google Scholar

[9]

Y. X. Li, Moser-Trudinger inequality on compact Riemannian manifolds of dimension two, J. Partial Differ. Equ., 14 (2001), 163-192.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

J. Peetre, Espaces d'interpolation et thereme de Soboleff, Ann. Inst. Fourier, 16 (1996), 279-317.   Google Scholar

[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.   Google Scholar

[22]

C. Tintarev, Trudinger-Moser inequality with remainder terms, J. Funct. Anal., 266 (2014), 55-66.  doi: 10.1016/j.jfa.2013.09.009.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

V. I. Yudovich, Some estimates connected with integral operators and with solutions of elliptic equations, Sov. Math. Docl., 2 (1961), 746-749.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[4]

W. Y. DingJ. JostJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

B. GidasW. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Commun. Math. Phys., 68 (1979), 209-243.   Google Scholar

[9]

Y. X. Li, Moser-Trudinger inequality on compact Riemannian manifolds of dimension two, J. Partial Differ. Equ., 14 (2001), 163-192.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

J. Peetre, Espaces d'interpolation et thereme de Soboleff, Ann. Inst. Fourier, 16 (1996), 279-317.   Google Scholar

[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.   Google Scholar

[22]

C. Tintarev, Trudinger-Moser inequality with remainder terms, J. Funct. Anal., 266 (2014), 55-66.  doi: 10.1016/j.jfa.2013.09.009.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

V. I. Yudovich, Some estimates connected with integral operators and with solutions of elliptic equations, Sov. Math. Docl., 2 (1961), 746-749.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Shiqiu Fu, Kanishka Perera. On a class of semipositone problems with singular Trudinger-Moser nonlinearities. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020452

[2]

Yifan Chen, Thomas Y. Hou. Function approximation via the subsampled Poincaré inequality. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 169-199. doi: 10.3934/dcds.2020296

[3]

Knut Hüper, Irina Markina, Fátima Silva Leite. A Lagrangian approach to extremal curves on Stiefel manifolds. Journal of Geometric Mechanics, 2020  doi: 10.3934/jgm.2020031

[4]

Xiaoxiao Li, Yingjing Shi, Rui Li, Shida Cao. Energy management method for an unpowered landing. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020180

[5]

Hao Wang. Uniform stability estimate for the Vlasov-Poisson-Boltzmann system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 657-680. doi: 10.3934/dcds.2020292

[6]

Shumin Li, Masahiro Yamamoto, Bernadette Miara. A Carleman estimate for the linear shallow shell equation and an inverse source problem. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 367-380. doi: 10.3934/dcds.2009.23.367

[7]

Yi Zhou, Jianli Liu. The initial-boundary value problem on a strip for the equation of time-like extremal surfaces. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 381-397. doi: 10.3934/dcds.2009.23.381

[8]

Xinfu Chen, Huiqiang Jiang, Guoqing Liu. Boundary spike of the singular limit of an energy minimizing problem. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3253-3290. doi: 10.3934/dcds.2020124

[9]

Marcello D'Abbicco, Giovanni Girardi, Giséle Ruiz Goldstein, Jerome A. Goldstein, Silvia Romanelli. Equipartition of energy for nonautonomous damped wave equations. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 597-613. doi: 10.3934/dcdss.2020364

[10]

Bahaaeldin Abdalla, Thabet Abdeljawad. Oscillation criteria for kernel function dependent fractional dynamic equations. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020443

[11]

Liping Tang, Ying Gao. Some properties of nonconvex oriented distance function and applications to vector optimization problems. Journal of Industrial & Management Optimization, 2021, 17 (1) : 485-500. doi: 10.3934/jimo.2020117

[12]

Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340

[13]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, 2021, 20 (1) : 449-465. doi: 10.3934/cpaa.2020276

[14]

Guillaume Cantin, M. A. Aziz-Alaoui. Dimension estimate of attractors for complex networks of reaction-diffusion systems applied to an ecological model. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020283

[15]

Luis Caffarelli, Fanghua Lin. Nonlocal heat flows preserving the L2 energy. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 49-64. doi: 10.3934/dcds.2009.23.49

[16]

Mohammed Abdulrazaq Kahya, Suhaib Abduljabbar Altamir, Zakariya Yahya Algamal. Improving whale optimization algorithm for feature selection with a time-varying transfer function. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 87-98. doi: 10.3934/naco.2020017

[17]

Lingfeng Li, Shousheng Luo, Xue-Cheng Tai, Jiang Yang. A new variational approach based on level-set function for convex hull problem with outliers. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020070

[18]

Juntao Sun, Tsung-fang Wu. The number of nodal solutions for the Schrödinger–Poisson system under the effect of the weight function. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021011

[19]

Madhurima Mukhopadhyay, Palash Sarkar, Shashank Singh, Emmanuel Thomé. New discrete logarithm computation for the medium prime case using the function field sieve. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020119

[20]

Kateřina Škardová, Tomáš Oberhuber, Jaroslav Tintěra, Radomír Chabiniok. Signed-distance function based non-rigid registration of image series with varying image intensity. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1145-1160. doi: 10.3934/dcdss.2020386

2019 Impact Factor: 1.105

Metrics

  • PDF downloads (118)
  • HTML views (85)
  • Cited by (0)

Other articles
by authors

[Back to Top]