Axisymmetry of locally bounded solutions to an Euler-Lagrange system of the weighted Hardy-Littlewood-Sobolev inequality

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May 2013, 33(5): 1987-2005. doi: 10.3934/dcds.2013.33.1987

Axisymmetry of locally bounded solutions to an Euler-Lagrange system of the weighted Hardy-Littlewood-Sobolev inequality

Yutian Lei ^1, and Zhongxue Lü ^2,

1.	Institute of Mathematics, School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210097, China
2.	School of Mathematical Sciences, Jiangsu Normal University, Xuzhou, 221116, China

Received November 2011 Revised March 2012 Published December 2012

Abstract
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This paper is concerned with the symmetry results for the $2k$-order singular Lane-Emden type partial differential system $$ \left\{\begin{array}{ll} (-\Delta)^k(|x|^{\alpha}u(x)) =|x|^{-\beta} v^{q}(x), \\ (-\Delta)^k(|x|^{\beta}v(x)) =|x|^{-\alpha} u^p(x), \end{array} \right. $$ and the weighted Hardy-Littlewood-Sobolev type integral system $$ \left \{ \begin{array}{l} u(x) = \frac{1}{|x|^{\alpha}}\int_{R^{n}} \frac{v^q(y)}{|y|^{\beta}|x-y|^{\lambda}} dy\\ v(x) = \frac{1}{|x|^{\beta}}\int_{R^{n}} \frac{u^p(y)}{|y|^{\alpha}|x-y|^{\lambda}} dy. \end{array} \right. $$ Here $x \in R^n \setminus \{0\}$. We first establish the equivalence of this integral system and an fractional order partial differential system, which includes the $2k$-order PDE system above. For the integral system, we prove that the positive locally bounded solutions are symmetric and decreasing about some axis by means of the method of moving planes in integral forms introduced by Chen-Li-Ou. In addition, we also show that the integrable solutions are locally bounded. Thus, the equivalence implies the positive solutions of the PDE system, particularly including the higher integer-order PDE system, also have the corresponding properties.

Keywords: weighted Hardy-Littlewood-Sobolev inequality, method of moving planes, axisymmetry., Higher-order Lane-Emden system.

Mathematics Subject Classification: 31B30, 35J48, 45E10, 45G0.

Citation: Yutian Lei, Zhongxue Lü. Axisymmetry of locally bounded solutions to an Euler-Lagrange system of the weighted Hardy-Littlewood-Sobolev inequality. Discrete and Continuous Dynamical Systems, 2013, 33 (5) : 1987-2005. doi: 10.3934/dcds.2013.33.1987

References:

[1]	L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math., 42 (1989), 271-297. doi: 10.1002/cpa.3160420304.
[2]	G. Caristi, L. D'Ambrosio and E. Mitidieri, Representation formulae for solutions to some classes of higher order systems and related Liouville theorems, Milan J. Math., 76 (2008), 27-67. doi: 10.1007/s00032-008-0090-3.
[3]	A. Chang and P. Yang, On uniqueness of an n-th order differential equation in conformal geometry, Math. Res. Letters, 4 (1997), 91-102.
[4]	W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622. doi: 10.1215/S0012-7094-91-06325-8.
[5]	W. Chen and C. Li, A priori estimates for prescribing scalar curvature equations, Ann. of Math., 145 (1997), 547-564. doi: 10.2307/2951844.
[6]	W. Chen and C. Li, The best constant in a weighted Hardy-Littlewood-Sobolev inequality, Proc. Amer. Math. Soc., 136 (2008), 955-962. doi: 10.1090/S0002-9939-07-09232-5.
[7]	W. Chen and C. Li, Super polyharmonic property of solutions for PDE systems and its applications, arXiv:1110.2539v1, 2011.
[8]	W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations, Comm. Partial Differential Equations, 30 (2005), 59-65. doi: 10.1081/PDE-200044445.
[9]	W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343. doi: 10.1002/cpa.20116.
[10]	Y. Fang and W. Chen, A Liouville type theorem for poly-harmonic Dirichlet problems in a half space, Adv. Math., 229 (2012), 2835-2867. doi: 10.1016/j.aim.2012.01.018.
[11]	B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n$, Collected in the book Mathematical\, Analysis and Applications, Which is 7a of the Book Series Advances in Mathematics. Supplementary Studies, Academic Press, New York, (1981).
[12]	D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer-Verlag, New York, 1977.
[13]	C. Jin and C. Li, Symmetry of solutions to some systems of integral equations, Proc. Amer. Math. Soc., 134 (2006), 1661-1670. doi: 10.1090/S0002-9939-05-08411-X.
[14]	C. Jin and C. Li, Qualitative analysis of some systems of integral equations, Calc. Var. Partial Differential Equations, 26 (2006), 447-457. doi: 10.1007/s00526-006-0013-5.
[15]	Y. Lei, C. Li and C. Ma, Decay estimation for positive solutions of a $\gamma$-Laplace equation, Discrete Contin. Dyn. Syst., 30 (2011), 547-558. doi: 10.3934/dcds.2011.30.547.
[16]	Y. Lei, C. Li and C. Ma, Asymptotic radial symmetry and growth estimates of positive solutions to weighted Hardy-Littlewood-Sobolev system, Calc. Var. Partial Differential Equations, 45 (2012), 43-61. doi: 10.1007/s00526-011-0450-7.
[17]	Y. Lei and C. Ma, Asymptotic behavior for solutions of some integral equations, Comm. Pure Appl. Anal., 10 (2011), 193-207. doi: 10.3934/cpaa.2011.10.193.
[18]	C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Invent. Math., 123 (1996), 221-231. doi: 10.1007/s002220050023.
[19]	C. Li and J. Lim, The singularity analysis of solutions to some integral equations, Comm. Pure Appl. Anal., 6 (2007), 453-464. doi: 10.3934/cpaa.2007.6.453.
[20]	Y. Li, Remark on some conformally invariant integral equations: The method of moving planes, J. Eur. Math. Soc., 6 (2004), 153-180.
[21]	E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118 (1983), 349-374. doi: 10.2307/2007032.
[22]	J. Liu, Y. Guo and Y. Zhang, Liouville-type theorems for polyharmonic systems in $R^n$, J. Differential equations, 225 (2006), 685-709. doi: 10.1016/j.jde.2005.10.016.
[23]	M. Onodera, On the shape of solutions to an integral system related to the weighted Hardy-Littlewood-Sobolev inequality, J. Math. Anal. Appl., 389 (2012), 498-510. doi: 10.1016/j.jmaa.2011.12.004.
[24]	J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318.
[25]	E. M. Stein, "Singular Integrals and Differentiability Properties of Function," Princetion Math. Series, 30. Princetion University Press, (1970), xiv+290 pp.
[26]	E. M. Stein and G. Weiss, Fractional integrals in $n$-dimensional Euclidean space, J. Math. Mech., 7 (1958), 503-514.
[27]	J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann., 313 (1999), 207-228. doi: 10.1007/s002080050258.

show all references

References:

[1]	L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math., 42 (1989), 271-297. doi: 10.1002/cpa.3160420304.
[2]	G. Caristi, L. D'Ambrosio and E. Mitidieri, Representation formulae for solutions to some classes of higher order systems and related Liouville theorems, Milan J. Math., 76 (2008), 27-67. doi: 10.1007/s00032-008-0090-3.
[3]	A. Chang and P. Yang, On uniqueness of an n-th order differential equation in conformal geometry, Math. Res. Letters, 4 (1997), 91-102.
[4]	W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622. doi: 10.1215/S0012-7094-91-06325-8.
[5]	W. Chen and C. Li, A priori estimates for prescribing scalar curvature equations, Ann. of Math., 145 (1997), 547-564. doi: 10.2307/2951844.
[6]	W. Chen and C. Li, The best constant in a weighted Hardy-Littlewood-Sobolev inequality, Proc. Amer. Math. Soc., 136 (2008), 955-962. doi: 10.1090/S0002-9939-07-09232-5.
[7]	W. Chen and C. Li, Super polyharmonic property of solutions for PDE systems and its applications, arXiv:1110.2539v1, 2011.
[8]	W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations, Comm. Partial Differential Equations, 30 (2005), 59-65. doi: 10.1081/PDE-200044445.
[9]	W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343. doi: 10.1002/cpa.20116.
[10]	Y. Fang and W. Chen, A Liouville type theorem for poly-harmonic Dirichlet problems in a half space, Adv. Math., 229 (2012), 2835-2867. doi: 10.1016/j.aim.2012.01.018.
[11]	B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n$, Collected in the book Mathematical\, Analysis and Applications, Which is 7a of the Book Series Advances in Mathematics. Supplementary Studies, Academic Press, New York, (1981).
[12]	D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer-Verlag, New York, 1977.
[13]	C. Jin and C. Li, Symmetry of solutions to some systems of integral equations, Proc. Amer. Math. Soc., 134 (2006), 1661-1670. doi: 10.1090/S0002-9939-05-08411-X.
[14]	C. Jin and C. Li, Qualitative analysis of some systems of integral equations, Calc. Var. Partial Differential Equations, 26 (2006), 447-457. doi: 10.1007/s00526-006-0013-5.
[15]	Y. Lei, C. Li and C. Ma, Decay estimation for positive solutions of a $\gamma$-Laplace equation, Discrete Contin. Dyn. Syst., 30 (2011), 547-558. doi: 10.3934/dcds.2011.30.547.
[16]	Y. Lei, C. Li and C. Ma, Asymptotic radial symmetry and growth estimates of positive solutions to weighted Hardy-Littlewood-Sobolev system, Calc. Var. Partial Differential Equations, 45 (2012), 43-61. doi: 10.1007/s00526-011-0450-7.
[17]	Y. Lei and C. Ma, Asymptotic behavior for solutions of some integral equations, Comm. Pure Appl. Anal., 10 (2011), 193-207. doi: 10.3934/cpaa.2011.10.193.
[18]	C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Invent. Math., 123 (1996), 221-231. doi: 10.1007/s002220050023.
[19]	C. Li and J. Lim, The singularity analysis of solutions to some integral equations, Comm. Pure Appl. Anal., 6 (2007), 453-464. doi: 10.3934/cpaa.2007.6.453.
[20]	Y. Li, Remark on some conformally invariant integral equations: The method of moving planes, J. Eur. Math. Soc., 6 (2004), 153-180.
[21]	E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118 (1983), 349-374. doi: 10.2307/2007032.
[22]	J. Liu, Y. Guo and Y. Zhang, Liouville-type theorems for polyharmonic systems in $R^n$, J. Differential equations, 225 (2006), 685-709. doi: 10.1016/j.jde.2005.10.016.
[23]	M. Onodera, On the shape of solutions to an integral system related to the weighted Hardy-Littlewood-Sobolev inequality, J. Math. Anal. Appl., 389 (2012), 498-510. doi: 10.1016/j.jmaa.2011.12.004.
[24]	J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318.
[25]	E. M. Stein, "Singular Integrals and Differentiability Properties of Function," Princetion Math. Series, 30. Princetion University Press, (1970), xiv+290 pp.
[26]	E. M. Stein and G. Weiss, Fractional integrals in $n$-dimensional Euclidean space, J. Math. Mech., 7 (1958), 503-514.
[27]	J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann., 313 (1999), 207-228. doi: 10.1007/s002080050258.

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