October  2013, 33(10): 4549-4566. doi: 10.3934/dcds.2013.33.4549

A Liouville theorem of degenerate elliptic equation and its application

1. 

School of Mathematical Science, Fudan University, Shanghai, 200433, China

Received  November 2012 Revised  February 2013 Published  April 2013

In this paper, we apply the moving plane method to the following degenerate elliptic equation arising from isometric embedding,\begin{equation*} yu_{yy}+au_y+\Delta_x u+u^\alpha=0\text{ in } \mathbb R^{n+1}_+,n\geq 1. \end{equation*} We get a Liouville theorem for subcritical case and classify the solutions for critical case. As an application, we derive the a priori bounds for positive solutions of some semi-linear degenerate elliptic equations.
Citation: Genggeng Huang. A Liouville theorem of degenerate elliptic equation and its application. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4549-4566. doi: 10.3934/dcds.2013.33.4549
References:
[1]

A. D. Alexandrov, Uniqueness theorems for surfaces in the large. V.,, Amer. Math. Soc. Transl.(2), 21 (1962), 412.   Google Scholar

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L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic with critical Sobolev growth,, Comm. Pure Appl. Math., 42 (1989), 271.  doi: 10.1002/cpa.3160420304.  Google Scholar

[3]

W.-X. Chen, C.-M. Li and B. OU, Classification of solutions for an integral equation,, Comm. Pure Appl. Math., 59 (2006), 330.  doi: 10.1002/cpa.20116.  Google Scholar

[4]

W.-X. Chen and C.-M. Li, Classification of positive solutions for nonlinear differential and integral systems with critical exponents,, Acta Math. Sci. Ser. B Engl. Ed., 29 (2009), 949.  doi: 10.1016/S0252-9602(09)60079-5.  Google Scholar

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W.-X. Chen and C.-M. Li, "Methods on Nonlinear Elliptic Equations,", AIMS, (2010).   Google Scholar

[6]

S-Y A. Chang and P. C. Yang, On uniqueness of solutions of n-th order differential equations in conformal geometry,, Math. Research Letters, 4 (1997), 91.   Google Scholar

[7]

G. Fichera, Sulle equazioni differenziali lineari ellittico-paraboliche del secondo ordine,, Atti Accad. Naz. Lincei. Mem. Cl. Sci. Fis. Mat. Nat. Sez. I(8), 5 (1956), 1.   Google Scholar

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G. Fichera, On a unified theory of boundary value problems for elliptic-parabolic equations of second order,, Boundary problems in differential equations, (1960), 97.   Google Scholar

[9]

B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry and related properties via maximum principle,, Comm. Math. Phys., 68 (1979), 209.   Google Scholar

[10]

B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbb R^n$,, Mathematical analysis and applications, (1981), 369.   Google Scholar

[11]

B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations,, Comm. Pure Appl. Math., 34 (1981), 525.  doi: 10.1002/cpa.3160340406.  Google Scholar

[12]

B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations,, Comm. Partial Differential Equations, 6 (1981), 883.  doi: 10.1080/03605308108820196.  Google Scholar

[13]

J.-X. Hong, On boundary value problems for mixed equations with characteristic degenerate surfaces,, Chin. Ann. of Math., 2 (1981), 407.   Google Scholar

[14]

J.-X. Hong and G.-G. Huang, $L^p$ and Hölder estimates for a class of degenerate elliptic partial differential equations and its applications,, Int. Math. Res. Notices, 2012 (): 2889.   Google Scholar

[15]

M. V. Keldyš, On certain cases of degeneration of equations of elliptic type on the boundary of a domain,, (Russian) Dokl. Akad. Nauk SSSR, 77 (1951), 181.   Google Scholar

[16]

C.-S. Lin, A classification of solutions of a conformally invariant fourth order equation in $\mathbb R^n$,, Commment. Math. Helv., 73 (1998), 206.  doi: 10.1007/s000140050052.  Google Scholar

[17]

C.-M. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations,, Invent. Math, 123 (1996), 221.  doi: 10.1007/s002220050023.  Google Scholar

[18]

C.-M. Li and L. Ma, Uniqueness of positive bound states to Schrödinger systems with critical exponents,, SIAM J. Math. Anal., 40 (2008), 1049.  doi: 10.1137/080712301.  Google Scholar

[19]

Y.-Y. Li, Remarks on some conformally invariant integral equations: The method of moving spheres,, J. Eur. Math. Soc., 6 (2004), 153.   Google Scholar

[20]

O. A. Oleinik and E. V. Radkevic, "Second Order Equations with Nonnegative Characteristic Form,", Translated from the Russian by Paul C. Fife. Plenum Press, (1973).   Google Scholar

[21]

J. Serrin, A symmetry problem in potential theory,, Arch. Rational Mech. Anal., 43 (1971), 304.   Google Scholar

[22]

J.-C. Wei and X.-W. Xu, Classification of solutions of higher order conformally invariant equations,, Math. Ann., 313 (1999), 207.  doi: 10.1007/s002080050258.  Google Scholar

[23]

X.-W. Xu, Classification of solutions of certain fourth-order nonlinear elliptic equations in $\mathbb R^4$,, Pacific J. Math., 225 (2006), 361.  doi: 10.2140/pjm.2006.225.361.  Google Scholar

show all references

References:
[1]

A. D. Alexandrov, Uniqueness theorems for surfaces in the large. V.,, Amer. Math. Soc. Transl.(2), 21 (1962), 412.   Google Scholar

[2]

L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic with critical Sobolev growth,, Comm. Pure Appl. Math., 42 (1989), 271.  doi: 10.1002/cpa.3160420304.  Google Scholar

[3]

W.-X. Chen, C.-M. Li and B. OU, Classification of solutions for an integral equation,, Comm. Pure Appl. Math., 59 (2006), 330.  doi: 10.1002/cpa.20116.  Google Scholar

[4]

W.-X. Chen and C.-M. Li, Classification of positive solutions for nonlinear differential and integral systems with critical exponents,, Acta Math. Sci. Ser. B Engl. Ed., 29 (2009), 949.  doi: 10.1016/S0252-9602(09)60079-5.  Google Scholar

[5]

W.-X. Chen and C.-M. Li, "Methods on Nonlinear Elliptic Equations,", AIMS, (2010).   Google Scholar

[6]

S-Y A. Chang and P. C. Yang, On uniqueness of solutions of n-th order differential equations in conformal geometry,, Math. Research Letters, 4 (1997), 91.   Google Scholar

[7]

G. Fichera, Sulle equazioni differenziali lineari ellittico-paraboliche del secondo ordine,, Atti Accad. Naz. Lincei. Mem. Cl. Sci. Fis. Mat. Nat. Sez. I(8), 5 (1956), 1.   Google Scholar

[8]

G. Fichera, On a unified theory of boundary value problems for elliptic-parabolic equations of second order,, Boundary problems in differential equations, (1960), 97.   Google Scholar

[9]

B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry and related properties via maximum principle,, Comm. Math. Phys., 68 (1979), 209.   Google Scholar

[10]

B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbb R^n$,, Mathematical analysis and applications, (1981), 369.   Google Scholar

[11]

B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations,, Comm. Pure Appl. Math., 34 (1981), 525.  doi: 10.1002/cpa.3160340406.  Google Scholar

[12]

B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations,, Comm. Partial Differential Equations, 6 (1981), 883.  doi: 10.1080/03605308108820196.  Google Scholar

[13]

J.-X. Hong, On boundary value problems for mixed equations with characteristic degenerate surfaces,, Chin. Ann. of Math., 2 (1981), 407.   Google Scholar

[14]

J.-X. Hong and G.-G. Huang, $L^p$ and Hölder estimates for a class of degenerate elliptic partial differential equations and its applications,, Int. Math. Res. Notices, 2012 (): 2889.   Google Scholar

[15]

M. V. Keldyš, On certain cases of degeneration of equations of elliptic type on the boundary of a domain,, (Russian) Dokl. Akad. Nauk SSSR, 77 (1951), 181.   Google Scholar

[16]

C.-S. Lin, A classification of solutions of a conformally invariant fourth order equation in $\mathbb R^n$,, Commment. Math. Helv., 73 (1998), 206.  doi: 10.1007/s000140050052.  Google Scholar

[17]

C.-M. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations,, Invent. Math, 123 (1996), 221.  doi: 10.1007/s002220050023.  Google Scholar

[18]

C.-M. Li and L. Ma, Uniqueness of positive bound states to Schrödinger systems with critical exponents,, SIAM J. Math. Anal., 40 (2008), 1049.  doi: 10.1137/080712301.  Google Scholar

[19]

Y.-Y. Li, Remarks on some conformally invariant integral equations: The method of moving spheres,, J. Eur. Math. Soc., 6 (2004), 153.   Google Scholar

[20]

O. A. Oleinik and E. V. Radkevic, "Second Order Equations with Nonnegative Characteristic Form,", Translated from the Russian by Paul C. Fife. Plenum Press, (1973).   Google Scholar

[21]

J. Serrin, A symmetry problem in potential theory,, Arch. Rational Mech. Anal., 43 (1971), 304.   Google Scholar

[22]

J.-C. Wei and X.-W. Xu, Classification of solutions of higher order conformally invariant equations,, Math. Ann., 313 (1999), 207.  doi: 10.1007/s002080050258.  Google Scholar

[23]

X.-W. Xu, Classification of solutions of certain fourth-order nonlinear elliptic equations in $\mathbb R^4$,, Pacific J. Math., 225 (2006), 361.  doi: 10.2140/pjm.2006.225.361.  Google Scholar

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