# American Institute of Mathematical Sciences

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

 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

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.
Citation: Yutian Lei, Zhongxue Lü. Axisymmetry of locally bounded solutions to an Euler-Lagrange system of the weighted Hardy-Littlewood-Sobolev inequality. Discrete & Continuous Dynamical Systems - A, 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. doi: 10.1002/cpa.3160420304. Google Scholar [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. doi: 10.1007/s00032-008-0090-3. Google Scholar [3] A. Chang and P. Yang, On uniqueness of an n-th order differential equation in conformal geometry,, Math. Res. Letters, 4 (1997), 91. Google Scholar [4] W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations,, Duke Math. J., 63 (1991), 615. doi: 10.1215/S0012-7094-91-06325-8. Google Scholar [5] W. Chen and C. Li, A priori estimates for prescribing scalar curvature equations,, Ann. of Math., 145 (1997), 547. doi: 10.2307/2951844. Google Scholar [6] W. Chen and C. Li, The best constant in a weighted Hardy-Littlewood-Sobolev inequality,, Proc. Amer. Math. Soc., 136 (2008), 955. doi: 10.1090/S0002-9939-07-09232-5. Google Scholar [7] W. Chen and C. Li, Super polyharmonic property of solutions for PDE systems and its applications,, arXiv:1110.2539v1, (2011). Google Scholar [8] W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations,, Comm. Partial Differential Equations, 30 (2005), 59. doi: 10.1081/PDE-200044445. Google Scholar [9] W. Chen, C. 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 [10] Y. Fang and W. Chen, A Liouville type theorem for poly-harmonic Dirichlet problems in a half space,, Adv. Math., 229 (2012), 2835. doi: 10.1016/j.aim.2012.01.018. Google Scholar [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\, 7a (1981). Google Scholar [12] D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Springer-Verlag, (1977). Google Scholar [13] C. Jin and C. Li, Symmetry of solutions to some systems of integral equations,, Proc. Amer. Math. Soc., 134 (2006), 1661. doi: 10.1090/S0002-9939-05-08411-X. Google Scholar [14] C. Jin and C. Li, Qualitative analysis of some systems of integral equations,, Calc. Var. Partial Differential Equations, 26 (2006), 447. doi: 10.1007/s00526-006-0013-5. Google Scholar [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. doi: 10.3934/dcds.2011.30.547. Google Scholar [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. doi: 10.1007/s00526-011-0450-7. Google Scholar [17] Y. Lei and C. Ma, Asymptotic behavior for solutions of some integral equations,, Comm. Pure Appl. Anal., 10 (2011), 193. doi: 10.3934/cpaa.2011.10.193. Google Scholar [18] C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations,, Invent. Math., 123 (1996), 221. doi: 10.1007/s002220050023. Google Scholar [19] C. Li and J. Lim, The singularity analysis of solutions to some integral equations,, Comm. Pure Appl. Anal., 6 (2007), 453. doi: 10.3934/cpaa.2007.6.453. Google Scholar [20] Y. Li, Remark on some conformally invariant integral equations: The method of moving planes,, J. Eur. Math. Soc., 6 (2004), 153. Google Scholar [21] E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities,, Ann. of Math., 118 (1983), 349. doi: 10.2307/2007032. Google Scholar [22] J. Liu, Y. Guo and Y. Zhang, Liouville-type theorems for polyharmonic systems in $R^n$,, J. Differential equations, 225 (2006), 685. doi: 10.1016/j.jde.2005.10.016. Google Scholar [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. doi: 10.1016/j.jmaa.2011.12.004. Google Scholar [24] J. Serrin, A symmetry problem in potential theory,, Arch. Rational Mech. Anal., 43 (1971), 304. Google Scholar [25] E. M. Stein, "Singular Integrals and Differentiability Properties of Function,", Princetion Math. Series, 30 (1970). Google Scholar [26] E. M. Stein and G. Weiss, Fractional integrals in $n$-dimensional Euclidean space,, J. Math. Mech., 7 (1958), 503. Google Scholar [27] J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations,, Math. Ann., 313 (1999), 207. doi: 10.1007/s002080050258. Google Scholar

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##### 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. doi: 10.1002/cpa.3160420304. Google Scholar [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. doi: 10.1007/s00032-008-0090-3. Google Scholar [3] A. Chang and P. Yang, On uniqueness of an n-th order differential equation in conformal geometry,, Math. Res. Letters, 4 (1997), 91. Google Scholar [4] W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations,, Duke Math. J., 63 (1991), 615. doi: 10.1215/S0012-7094-91-06325-8. Google Scholar [5] W. Chen and C. Li, A priori estimates for prescribing scalar curvature equations,, Ann. of Math., 145 (1997), 547. doi: 10.2307/2951844. Google Scholar [6] W. Chen and C. Li, The best constant in a weighted Hardy-Littlewood-Sobolev inequality,, Proc. Amer. Math. Soc., 136 (2008), 955. doi: 10.1090/S0002-9939-07-09232-5. Google Scholar [7] W. Chen and C. Li, Super polyharmonic property of solutions for PDE systems and its applications,, arXiv:1110.2539v1, (2011). Google Scholar [8] W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations,, Comm. Partial Differential Equations, 30 (2005), 59. doi: 10.1081/PDE-200044445. Google Scholar [9] W. Chen, C. 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 [10] Y. Fang and W. Chen, A Liouville type theorem for poly-harmonic Dirichlet problems in a half space,, Adv. Math., 229 (2012), 2835. doi: 10.1016/j.aim.2012.01.018. Google Scholar [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\, 7a (1981). Google Scholar [12] D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Springer-Verlag, (1977). Google Scholar [13] C. Jin and C. Li, Symmetry of solutions to some systems of integral equations,, Proc. Amer. Math. Soc., 134 (2006), 1661. doi: 10.1090/S0002-9939-05-08411-X. Google Scholar [14] C. Jin and C. Li, Qualitative analysis of some systems of integral equations,, Calc. Var. Partial Differential Equations, 26 (2006), 447. doi: 10.1007/s00526-006-0013-5. Google Scholar [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. doi: 10.3934/dcds.2011.30.547. Google Scholar [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. doi: 10.1007/s00526-011-0450-7. Google Scholar [17] Y. Lei and C. Ma, Asymptotic behavior for solutions of some integral equations,, Comm. Pure Appl. Anal., 10 (2011), 193. doi: 10.3934/cpaa.2011.10.193. Google Scholar [18] C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations,, Invent. Math., 123 (1996), 221. doi: 10.1007/s002220050023. Google Scholar [19] C. Li and J. Lim, The singularity analysis of solutions to some integral equations,, Comm. Pure Appl. Anal., 6 (2007), 453. doi: 10.3934/cpaa.2007.6.453. Google Scholar [20] Y. Li, Remark on some conformally invariant integral equations: The method of moving planes,, J. Eur. Math. Soc., 6 (2004), 153. Google Scholar [21] E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities,, Ann. of Math., 118 (1983), 349. doi: 10.2307/2007032. Google Scholar [22] J. Liu, Y. Guo and Y. Zhang, Liouville-type theorems for polyharmonic systems in $R^n$,, J. Differential equations, 225 (2006), 685. doi: 10.1016/j.jde.2005.10.016. Google Scholar [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. doi: 10.1016/j.jmaa.2011.12.004. Google Scholar [24] J. Serrin, A symmetry problem in potential theory,, Arch. Rational Mech. Anal., 43 (1971), 304. Google Scholar [25] E. M. Stein, "Singular Integrals and Differentiability Properties of Function,", Princetion Math. Series, 30 (1970). Google Scholar [26] E. M. Stein and G. Weiss, Fractional integrals in $n$-dimensional Euclidean space,, J. Math. Mech., 7 (1958), 503. Google Scholar [27] J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations,, Math. Ann., 313 (1999), 207. doi: 10.1007/s002080050258. Google Scholar
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