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A curve of positive solutions for an indefinite sublinear Dirichlet problem
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 |
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 [
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. 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. |
[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. |
[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. 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. |
[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 Figueiredo, J. M. DO |
[13] |
J. Heinonen, T. 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 |
[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. 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. |
[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. |
[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. 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. |
[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 Figueiredo, J. M. DO |
[13] |
J. Heinonen, T. 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 |
[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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