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

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

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].

Citation: Changliang Zhou, Chunqin Zhou. On the anisotropic Moser-Trudinger inequality for unbounded domains in $ \mathbb R^{n} $. Discrete and 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.

[2]

A. AlvinoV. FeroneG. 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.

[3]

M. BelloniV. 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.

[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.

[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.

[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. 

[7]

R. CernyA. 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.

[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.

[9]

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

[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.

[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.

[12]

D. G. DE FigueiredoJ. 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.

[13] J. HeinonenT. Kilpelainen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford University Press, 1993. 
[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.

[15]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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]

J. M. Do ÒE. 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.

[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. 

[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.

[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.

[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.

[30]

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

[31]

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

[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.

[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.

[34]

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

[35]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[2]

A. AlvinoV. FeroneG. 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.

[3]

M. BelloniV. 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.

[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.

[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.

[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. 

[7]

R. CernyA. 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.

[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.

[9]

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

[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.

[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.

[12]

D. G. DE FigueiredoJ. 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.

[13] J. HeinonenT. Kilpelainen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford University Press, 1993. 
[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.

[15]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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]

J. M. Do ÒE. 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.

[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. 

[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.

[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.

[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.

[30]

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

[31]

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

[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.

[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.

[34]

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

[35]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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