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February  2020, 40(2): 847-881. doi: 10.3934/dcds.2020064

On the anisotropic Moser-Trudinger inequality for unbounded domains in $ \mathbb R^{n} $

Changliang Zhou 1,, and  Chunqin Zhou 2,,

1. 

School of Science, East China University of Technology, Nanchang, China
2. 

School of Mathematical Sciences, MOE-LSC, Shanghai Jiao Tong University, Shanghai, China

* Corresponding author: Chunqin Zhou and Changliang Zhou

Received  February 2019 Revised  August 2019 Published  November 2019

Fund Project: The authors are supported partially by NSFC of China (No. 11771285). The first author is also supported by the funding for the Doctoral Research of ECUT under grant No. DHBK2018053.

In this paper, we investigate a sharp Moser-Trudinger inequality which involves the anisotropic Sobolev norm in unbounded domains. Under this anisotropic Sobolev norm, we establish the Lions type concentration-compactness alternative firstly. Then by using a blow-up procedure, we obtain the existence of extremal functions for this sharp geometric inequality. In particular, we combine the low dimension case of $ n = 2 $ and the high dimension case of $ n\geq 3 $ to prove the existence of the extremal functions, which is different from the arguments of isotropic case, see [5,19].

Keywords: Moser-Trudinger inequality, anisotropic Sobolev norm, blow-up analysis, existence of extremal functions.
Mathematics Subject Classification: 35A05, 35J65.
Citation: Changliang Zhou, Chunqin Zhou. On the anisotropic Moser-Trudinger inequality for unbounded domains in $ \mathbb R^{n} $. Discrete & Continuous Dynamical Systems, 2020, 40 (2) : 847-881. doi: 10.3934/dcds.2020064
References:
[1]

Adimurthi and K. Sandeep, A Singular Moser-Trudinger embedding and its applications, NoDEA Nolinear Differential Equations Appl., 13 (2007), 585-603.  doi: 10.1007/s00030-006-4025-9.  Google Scholar

[2]

A. Alvino, V. Ferone, G. Trombetti and P. L. Lions, convex symmetrization and applications, Ann. Inst.H. Poincare. Anal. Nonlineaire, 14 (1997), 275-293.  doi: 10.1016/S0294-1449(97)80147-3.  Google Scholar

[3]

M. Belloni, V. Ferone and B. Kawohl, Isoperimetric inequalities, wulffshape and related questions for strongly nonlinear elliptic operators, Z. Angew. Math. Phys., 54 (2003), 771-783.  doi: 10.1007/s00033-003-3209-y.  Google Scholar

[4]

G. Bellettini and M. Paolini, Anisotropic motion by mean curvature in the context of Finsler geometry, Hokkaido Math. J., 25 (1996), 537-566.  doi: 10.14492/hokmj/1351516749.  Google Scholar

[5]

B. Ruf, A sharp Trudinger-Moser inequality for unbounded domains in $\mathbb R^{2}$, J. Funct. Anal., 219 (2005), 340-367.  doi: 10.1016/j.jfa.2004.06.013.  Google Scholar

[6]

L. Carleson and S. Y. A. Chang, On the existence of an extremal function for an inequality of J. Moser, Bull. Sci. Math., 100 (1986), 113-127.   Google Scholar

[7]

R. Cerny, A. Cianchi and S. Henel, Concentration-compactness principles for Moser-Trudinger inequalities:new results and proofs, Ann. Mat. Pura. Appl., 192 (2013), 225-243.  doi: 10.1007/s10231-011-0220-3.  Google Scholar

[8]

L. Damascelli, Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results, Ann. Inst. H.Poincare Anal. Non Lineaire, 15 (1998), 493-516.  doi: 10.1016/S0294-1449(98)80032-2.  Google Scholar

[9]

M. Flucher, Extremal functions of for the Trudinger-Moser inequality in two dimensions, Comm. Math. Helv., 67 (1992), 471-497.   Google Scholar

[10]

V. Ferone and B. Kawohl, Remarks on a Finsler-Laplacian, Proc. Amer. Math. Soc., 137 (2009), 247-253.  doi: 10.1090/S0002-9939-08-09554-3.  Google Scholar

[11]

I. Fonseca and S. Muller, A uniqueness proof for the Wulff theorem, Proc. Roy. Soc. Edinburgh Scet., 119 (1991), 125-136.  doi: 10.1017/S0308210500028365.  Google Scholar

[12]

D. G. DE Figueiredo, J. M. DO Ò 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

[13] J. Heinonen, T. Kilpelainen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford University Press, 1993.   Google Scholar
[14]

M. Ishiwata, Existence and nonexistence of maximizers for variational problems associated with Trudinger-Moser type inequalities in $\mathbb R^{n}$, Math. Ann., 351 (2011), 781-804.  doi: 10.1007/s00208-010-0618-z.  Google Scholar

[15]

S. Kichenassamy and L. Veron, Singular Solutions of the p Laplace Equation, Math. Ann., 275 (1986), 599-615.  doi: 10.1007/BF01459140.  Google Scholar

[16]

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

[17]

G. Z. Lu and Y. Y. Li, Sharp constant and extremal functiion for the improved Moser-Trudinger inequality involving Lp norm in two dimension, Discrete Contin. Dyn. Syst., 25 (2009), 963-979.  doi: 10.3934/dcds.2009.25.963.  Google Scholar

[18]

Y. X. Li, Extremal functions for the Moser-Trudinger inequalities on compact Riemannian manifolds, Ser. A. Mathematics, 48 (2005), 618-648.  doi: 10.1360/04ys0050.  Google Scholar

[19]

Y. X. Li and B. Ruf, A shape Trudinger-Moser type inequality for unbounded domains in $\mathbb{R}^{n}$, Indiana Univ. Math. J., 57 (2008), 451-480.  doi: 10.1512/iumj.2008.57.3137.  Google Scholar

[20]

G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal. T. M. A., 12 (1988), 1203-1219.  doi: 10.1016/0362-546X(88)90053-3.  Google Scholar

[21]

N. Lam and G. Z. Lu, Existence and multiplicity of solutions to equations of n-Laplacian type with critical exponential growth in $\mathbb R^{n}$, J. Funct. Anal., 262 (2012), 1132-1165.  doi: 10.1016/j.jfa.2011.10.012.  Google Scholar

[22]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The limit case, art 1, Rev. Mat. Iberoamericana, 1 (1985), 145-201.  doi: 10.4171/RMI/6.  Google Scholar

[23]

G. Z. Lu and M. C. Zhu, A sharp Trudinger-Moser type inequality involving Ln norm in the entire space $\mathbb R^{n}$, J. Differential Equations, 267 (2019), 3046-3082.  doi: 10.1016/j.jde.2019.03.037.  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]

J. M. Do Ò, E. Medeiros and U. Severo, On a quasilinear nonhomogeneneous elliptic equation with critical growth in $\mathbb R^{n}$, J. Differential Equations, 246 (2009), 1363-1386.  doi: 10.1016/j.jde.2008.11.020.  Google Scholar

[26]

S. Pohozaev, The sobolev embedding in the special case pl=n, Proceedings of the Technical Scientific Conference on Advances of Scientific Reseach.Mathmatic sections, Mosco. Energet. inst., 11 (1965), 158-170.   Google Scholar

[27]

F. D. Pietra and G. D. Blasio, Blow-up solutions for some nonlinear elliptic equations involving a Finsler-Laplacian, Publicacions matematiques, 61 (2017), 213-228.  doi: 10.5565/PUBLMAT_61117_08.  Google Scholar

[28]

M. Struwe, Positive solution of critical semilinear elliptic equations on non-contractible planar domain, J. Eur. Math.Soc., 2 (2000), 329-388.  doi: 10.1007/s100970000023.  Google Scholar

[29]

S. Adachi and K. Tanaka, Trudinger type inequalities in $\mathbb{R}^{n}$ and their best exponent, Proc. Amer. Math. Soc., 128 (2000), 2051-2057.  doi: 10.1090/S0002-9939-99-05180-1.  Google Scholar

[30]

J. Serrin, Local behavior of solutions of quasi-linear equations, Acta. Math., 111 (1964), 247-302.  doi: 10.1007/BF02391014.  Google Scholar

[31]

J. Serrin, Isolated singularities of solutions quasilinear equations, Acta. Math., 113 (1965), 219-240.  doi: 10.1007/BF02391778.  Google Scholar

[32]

N. S. Trudinger, On embedding into Orlicz space and some applications, J. Math. Mech., 17 (1967), 473-484.  doi: 10.1512/iumj.1968.17.17028.  Google Scholar

[33]

P. Tolksdorf, Regularity for a more general class of qusilinear elliptic equations, J. Differential Equations, 51 (1984), 126-150.  doi: 10.1016/0022-0396(84)90105-0.  Google Scholar

[34]

G. Talenti, Elliptic equations and rearrangements, Ann. Sc. Norm. Super. Pisa. Ci. Sci., 3 (1976), 697-718.   Google Scholar

[35]

W. Beckner, Estimates on Moser embedding, Potential Anal., 20 (2004), 345-359.  doi: 10.1023/B:POTA.0000009813.38619.47.  Google Scholar

[36]

G. F. Wang and C. Xia, Blow-up analysis of a Finsler-Liouville equation in two dimensions, J. Differential. Equations, 252 (2012), 1668-1700.  doi: 10.1016/j.jde.2011.08.001.  Google Scholar

[37]

G. F. Wang and C. Xia, An optimal anisotropic poincare inequality for convex domains, Pacific Journal of Mathematics, 258 (2012), 305-325.  doi: 10.2140/pjm.2012.258.305.  Google Scholar

[38]

G. F. Wang and C. Xia, A characterization of the wuff shape by an overdetermined anisotropic PDE, Arch. Ration. Mech. Anal., 99 (2011), 99-115.  doi: 10.1007/s00205-010-0323-9.  Google Scholar

[39]

G. F. Wang and D. Ye, A hardy-Moser-Trudinger inequality, Advances in Mathmatics, 230 (2012), 294-320.  doi: 10.1016/j.aim.2011.12.001.  Google Scholar

[40]

R. L. Xie and H. J. Gong, A priori estimates and blow-up behavior for solutions of $-Q_{n}u=Ve^{u}$ in bounded domain in $\mathbb{R}^{n}$, Science China Mathematics, 59 (2016), 479-492.  doi: 10.1007/s11425-015-5060-y.  Google Scholar

[41]

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

[42]

J. Y. Zhu, Improved Moser-Trudinger inequality involving Lp norm in n dimensions, Advanced Nonlinear Studies, 14 (2014), 273-293.  doi: 10.1515/ans-2014-0202.  Google Scholar

[43]

C. L. Zhou and C. Q. Zhou, Moser-Trudinger inequality involving the anisotropic dirichlet Norm $(\int_{\Omega}F^{n}(\nabla u)dx)^{\frac{1}{n}}$ on $W_{0}^{1,n}(\Omega)$, J. Funct. Anal., 276 (2019), 2901-2935.  doi: 10.1016/j.jfa.2018.12.001.  Google Scholar

show all references

References:
[1]

Adimurthi and K. Sandeep, A Singular Moser-Trudinger embedding and its applications, NoDEA Nolinear Differential Equations Appl., 13 (2007), 585-603.  doi: 10.1007/s00030-006-4025-9.  Google Scholar

[2]

A. Alvino, V. Ferone, G. Trombetti and P. L. Lions, convex symmetrization and applications, Ann. Inst.H. Poincare. Anal. Nonlineaire, 14 (1997), 275-293.  doi: 10.1016/S0294-1449(97)80147-3.  Google Scholar

[3]

M. Belloni, V. Ferone and B. Kawohl, Isoperimetric inequalities, wulffshape and related questions for strongly nonlinear elliptic operators, Z. Angew. Math. Phys., 54 (2003), 771-783.  doi: 10.1007/s00033-003-3209-y.  Google Scholar

[4]

G. Bellettini and M. Paolini, Anisotropic motion by mean curvature in the context of Finsler geometry, Hokkaido Math. J., 25 (1996), 537-566.  doi: 10.14492/hokmj/1351516749.  Google Scholar

[5]

B. Ruf, A sharp Trudinger-Moser inequality for unbounded domains in $\mathbb R^{2}$, J. Funct. Anal., 219 (2005), 340-367.  doi: 10.1016/j.jfa.2004.06.013.  Google Scholar

[6]

L. Carleson and S. Y. A. Chang, On the existence of an extremal function for an inequality of J. Moser, Bull. Sci. Math., 100 (1986), 113-127.   Google Scholar

[7]

R. Cerny, A. Cianchi and S. Henel, Concentration-compactness principles for Moser-Trudinger inequalities:new results and proofs, Ann. Mat. Pura. Appl., 192 (2013), 225-243.  doi: 10.1007/s10231-011-0220-3.  Google Scholar

[8]

L. Damascelli, Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results, Ann. Inst. H.Poincare Anal. Non Lineaire, 15 (1998), 493-516.  doi: 10.1016/S0294-1449(98)80032-2.  Google Scholar

[9]

M. Flucher, Extremal functions of for the Trudinger-Moser inequality in two dimensions, Comm. Math. Helv., 67 (1992), 471-497.   Google Scholar

[10]

V. Ferone and B. Kawohl, Remarks on a Finsler-Laplacian, Proc. Amer. Math. Soc., 137 (2009), 247-253.  doi: 10.1090/S0002-9939-08-09554-3.  Google Scholar

[11]

I. Fonseca and S. Muller, A uniqueness proof for the Wulff theorem, Proc. Roy. Soc. Edinburgh Scet., 119 (1991), 125-136.  doi: 10.1017/S0308210500028365.  Google Scholar

[12]

D. G. DE Figueiredo, J. M. DO Ò 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

[13] J. Heinonen, T. Kilpelainen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford University Press, 1993.   Google Scholar
[14]

M. Ishiwata, Existence and nonexistence of maximizers for variational problems associated with Trudinger-Moser type inequalities in $\mathbb R^{n}$, Math. Ann., 351 (2011), 781-804.  doi: 10.1007/s00208-010-0618-z.  Google Scholar

[15]

S. Kichenassamy and L. Veron, Singular Solutions of the p Laplace Equation, Math. Ann., 275 (1986), 599-615.  doi: 10.1007/BF01459140.  Google Scholar

[16]

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

[17]

G. Z. Lu and Y. Y. Li, Sharp constant and extremal functiion for the improved Moser-Trudinger inequality involving Lp norm in two dimension, Discrete Contin. Dyn. Syst., 25 (2009), 963-979.  doi: 10.3934/dcds.2009.25.963.  Google Scholar

[18]

Y. X. Li, Extremal functions for the Moser-Trudinger inequalities on compact Riemannian manifolds, Ser. A. Mathematics, 48 (2005), 618-648.  doi: 10.1360/04ys0050.  Google Scholar

[19]

Y. X. Li and B. Ruf, A shape Trudinger-Moser type inequality for unbounded domains in $\mathbb{R}^{n}$, Indiana Univ. Math. J., 57 (2008), 451-480.  doi: 10.1512/iumj.2008.57.3137.  Google Scholar

[20]

G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal. T. M. A., 12 (1988), 1203-1219.  doi: 10.1016/0362-546X(88)90053-3.  Google Scholar

[21]

N. Lam and G. Z. Lu, Existence and multiplicity of solutions to equations of n-Laplacian type with critical exponential growth in $\mathbb R^{n}$, J. Funct. Anal., 262 (2012), 1132-1165.  doi: 10.1016/j.jfa.2011.10.012.  Google Scholar

[22]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The limit case, art 1, Rev. Mat. Iberoamericana, 1 (1985), 145-201.  doi: 10.4171/RMI/6.  Google Scholar

[23]

G. Z. Lu and M. C. Zhu, A sharp Trudinger-Moser type inequality involving Ln norm in the entire space $\mathbb R^{n}$, J. Differential Equations, 267 (2019), 3046-3082.  doi: 10.1016/j.jde.2019.03.037.  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]

J. M. Do Ò, E. Medeiros and U. Severo, On a quasilinear nonhomogeneneous elliptic equation with critical growth in $\mathbb R^{n}$, J. Differential Equations, 246 (2009), 1363-1386.  doi: 10.1016/j.jde.2008.11.020.  Google Scholar

[26]

S. Pohozaev, The sobolev embedding in the special case pl=n, Proceedings of the Technical Scientific Conference on Advances of Scientific Reseach.Mathmatic sections, Mosco. Energet. inst., 11 (1965), 158-170.   Google Scholar

[27]

F. D. Pietra and G. D. Blasio, Blow-up solutions for some nonlinear elliptic equations involving a Finsler-Laplacian, Publicacions matematiques, 61 (2017), 213-228.  doi: 10.5565/PUBLMAT_61117_08.  Google Scholar

[28]

M. Struwe, Positive solution of critical semilinear elliptic equations on non-contractible planar domain, J. Eur. Math.Soc., 2 (2000), 329-388.  doi: 10.1007/s100970000023.  Google Scholar

[29]

S. Adachi and K. Tanaka, Trudinger type inequalities in $\mathbb{R}^{n}$ and their best exponent, Proc. Amer. Math. Soc., 128 (2000), 2051-2057.  doi: 10.1090/S0002-9939-99-05180-1.  Google Scholar

[30]

J. Serrin, Local behavior of solutions of quasi-linear equations, Acta. Math., 111 (1964), 247-302.  doi: 10.1007/BF02391014.  Google Scholar

[31]

J. Serrin, Isolated singularities of solutions quasilinear equations, Acta. Math., 113 (1965), 219-240.  doi: 10.1007/BF02391778.  Google Scholar

[32]

N. S. Trudinger, On embedding into Orlicz space and some applications, J. Math. Mech., 17 (1967), 473-484.  doi: 10.1512/iumj.1968.17.17028.  Google Scholar

[33]

P. Tolksdorf, Regularity for a more general class of qusilinear elliptic equations, J. Differential Equations, 51 (1984), 126-150.  doi: 10.1016/0022-0396(84)90105-0.  Google Scholar

[34]

G. Talenti, Elliptic equations and rearrangements, Ann. Sc. Norm. Super. Pisa. Ci. Sci., 3 (1976), 697-718.   Google Scholar

[35]

W. Beckner, Estimates on Moser embedding, Potential Anal., 20 (2004), 345-359.  doi: 10.1023/B:POTA.0000009813.38619.47.  Google Scholar

[36]

G. F. Wang and C. Xia, Blow-up analysis of a Finsler-Liouville equation in two dimensions, J. Differential. Equations, 252 (2012), 1668-1700.  doi: 10.1016/j.jde.2011.08.001.  Google Scholar

[37]

G. F. Wang and C. Xia, An optimal anisotropic poincare inequality for convex domains, Pacific Journal of Mathematics, 258 (2012), 305-325.  doi: 10.2140/pjm.2012.258.305.  Google Scholar

[38]

G. F. Wang and C. Xia, A characterization of the wuff shape by an overdetermined anisotropic PDE, Arch. Ration. Mech. Anal., 99 (2011), 99-115.  doi: 10.1007/s00205-010-0323-9.  Google Scholar

[39]

G. F. Wang and D. Ye, A hardy-Moser-Trudinger inequality, Advances in Mathmatics, 230 (2012), 294-320.  doi: 10.1016/j.aim.2011.12.001.  Google Scholar

[40]

R. L. Xie and H. J. Gong, A priori estimates and blow-up behavior for solutions of $-Q_{n}u=Ve^{u}$ in bounded domain in $\mathbb{R}^{n}$, Science China Mathematics, 59 (2016), 479-492.  doi: 10.1007/s11425-015-5060-y.  Google Scholar

[41]

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

[42]

J. Y. Zhu, Improved Moser-Trudinger inequality involving Lp norm in n dimensions, Advanced Nonlinear Studies, 14 (2014), 273-293.  doi: 10.1515/ans-2014-0202.  Google Scholar

[43]

C. L. Zhou and C. Q. Zhou, Moser-Trudinger inequality involving the anisotropic dirichlet Norm $(\int_{\Omega}F^{n}(\nabla u)dx)^{\frac{1}{n}}$ on $W_{0}^{1,n}(\Omega)$, J. Funct. Anal., 276 (2019), 2901-2935.  doi: 10.1016/j.jfa.2018.12.001.  Google Scholar

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