A Liouville theorem of degenerate elliptic equation and its application

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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

Abstract
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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.

Keywords: blow-up, semi-linear, Liouville theorem, degenerate., moving plane.

Mathematics Subject Classification: Primary: 35J61, 35J70; Secondary: 35B53, 35B4.

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:

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[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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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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