# American Institute of Mathematical Sciences

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 and Continuous Dynamical Systems, 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), 412-416. [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), 271-297. doi: 10.1002/cpa.3160420304. [3] W.-X. Chen, C.-M. Li and B. OU, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343. doi: 10.1002/cpa.20116. [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), 949-960. doi: 10.1016/S0252-9602(09)60079-5. [5] W.-X. Chen and C.-M. Li, "Methods on Nonlinear Elliptic Equations," AIMS, 2010. [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), 91-102. [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), 1-30. [8] G. Fichera, On a unified theory of boundary value problems for elliptic-parabolic equations of second order, Boundary problems in differential equations, Univ. of Wisconsin Press, Madison, Wis., (1960), 97-120. [9] B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry and related properties via maximum principle, Comm. Math. Phys., 68 (1979), 209-243. [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, Part A, 369-402. Advances in Mathematics, Supplementary Studies, 7a. Academic Press, New York-London, (1981). [11] B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34 (1981), 525-598. doi: 10.1002/cpa.3160340406. [12] B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations, 6 (1981), 883-901. doi: 10.1080/03605308108820196. [13] J.-X. Hong, On boundary value problems for mixed equations with characteristic degenerate surfaces, Chin. Ann. of Math., 2 (1981), 407-424. [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, 2889-2941. [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), 181-183. [16] C.-S. Lin, A classification of solutions of a conformally invariant fourth order equation in $\mathbb R^n$, Commment. Math. Helv., 73 (1998), 206-231. doi: 10.1007/s000140050052. [17] C.-M. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Invent. Math, 123 (1996), 221-231. doi: 10.1007/s002220050023. [18] C.-M. Li and L. Ma, Uniqueness of positive bound states to Schrödinger systems with critical exponents, SIAM J. Math. Anal., 40 (2008), 1049-1057. doi: 10.1137/080712301. [19] Y.-Y. Li, Remarks on some conformally invariant integral equations: The method of moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180. [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, New York-London, 1973. [21] J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318. [22] J.-C. Wei and X.-W. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann., 313 (1999), 207-228. doi: 10.1007/s002080050258. [23] X.-W. Xu, Classification of solutions of certain fourth-order nonlinear elliptic equations in $\mathbb R^4$, Pacific J. Math., 225 (2006), 361-378. doi: 10.2140/pjm.2006.225.361.

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##### References:
 [1] A. D. Alexandrov, Uniqueness theorems for surfaces in the large. V., Amer. Math. Soc. Transl.(2), 21 (1962), 412-416. [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), 271-297. doi: 10.1002/cpa.3160420304. [3] W.-X. Chen, C.-M. Li and B. OU, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343. doi: 10.1002/cpa.20116. [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), 949-960. doi: 10.1016/S0252-9602(09)60079-5. [5] W.-X. Chen and C.-M. Li, "Methods on Nonlinear Elliptic Equations," AIMS, 2010. [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), 91-102. [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), 1-30. [8] G. Fichera, On a unified theory of boundary value problems for elliptic-parabolic equations of second order, Boundary problems in differential equations, Univ. of Wisconsin Press, Madison, Wis., (1960), 97-120. [9] B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry and related properties via maximum principle, Comm. Math. Phys., 68 (1979), 209-243. [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, Part A, 369-402. Advances in Mathematics, Supplementary Studies, 7a. Academic Press, New York-London, (1981). [11] B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34 (1981), 525-598. doi: 10.1002/cpa.3160340406. [12] B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations, 6 (1981), 883-901. doi: 10.1080/03605308108820196. [13] J.-X. Hong, On boundary value problems for mixed equations with characteristic degenerate surfaces, Chin. Ann. of Math., 2 (1981), 407-424. [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, 2889-2941. [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), 181-183. [16] C.-S. Lin, A classification of solutions of a conformally invariant fourth order equation in $\mathbb R^n$, Commment. Math. Helv., 73 (1998), 206-231. doi: 10.1007/s000140050052. [17] C.-M. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Invent. Math, 123 (1996), 221-231. doi: 10.1007/s002220050023. [18] C.-M. Li and L. Ma, Uniqueness of positive bound states to Schrödinger systems with critical exponents, SIAM J. Math. Anal., 40 (2008), 1049-1057. doi: 10.1137/080712301. [19] Y.-Y. Li, Remarks on some conformally invariant integral equations: The method of moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180. [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, New York-London, 1973. [21] J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318. [22] J.-C. Wei and X.-W. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann., 313 (1999), 207-228. doi: 10.1007/s002080050258. [23] X.-W. Xu, Classification of solutions of certain fourth-order nonlinear elliptic equations in $\mathbb R^4$, Pacific J. Math., 225 (2006), 361-378. doi: 10.2140/pjm.2006.225.361.
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