American Institute of Mathematical Sciences

November  2011, 30(4): 1083-1093. doi: 10.3934/dcds.2011.30.1083

Radial symmetry of solutions for some integral systems of Wolff type

 1 Department of Mathematics, Yeshiva University, New York, NY 10033 2 Department of Applied Mathematics, University of Colorado at Boulder

Received  March 2010 Revised  August 2010 Published  May 2011

We consider the fully nonlinear integral systems involving Wolff potentials:

$\u(x) = W_{\beta, \gamma}(v^q)(x)$, $\x \in R^n$;
$\v(x) = W_{\beta, \gamma} (u^p)(x)$, $\x \in R^n$;

(1)

where

$\W_{\beta,\gamma} (f)(x) = \int_0^{\infty}$ $[ \frac{\int_{B_t(x)} f(y) dy}{t^{n-\beta\gamma}} ]^{\frac{1}{\gamma-1}} \frac{d t}{t}.$

After modifying and refining our techniques on the method of moving planes in integral forms, we obtain radial symmetry and monotonicity for the positive solutions to systems (1).
This system includes many known systems as special cases, in particular, when $\beta = \frac{\alpha}{2}$ and $\gamma = 2$, system (1) reduces to

$\u(x) = \int_{R^{n}} \frac{1}{|x-y|^{n-\alpha}} v(y)^q dy$, $\ x \in R^n$,
$v(x) = \int_{R^{n}} \frac{1}{|x-y|^{n-\alpha}} u(y)^p dy$, $\ x \in R^n$.

(2)

The solutions $(u,v)$ of (2) are critical points of the functional associated with the well-known Hardy-Littlewood-Sobolev inequality. We can show that (2) is equivalent to a system of semi-linear elliptic PDEs

$(-\Delta)^{\alpha/2} u = v^q$, in $R^n$,
$(-\Delta)^{\alpha/2} v = u^p$, in $R^n$

(3)

which comprises the well-known Lane-Emden system and Yamabe equation.
Citation: Wenxiong Chen, Congming Li. Radial symmetry of solutions for some integral systems of Wolff type. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1083-1093. doi: 10.3934/dcds.2011.30.1083
References:
 [1] L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth,, C. P. A. M., XLII (1989), 271. [2] W. Chen, C. Jin, C. Li and C. Lim, Weighted Hardy-Littlewood-Sobolev inequalities and system of integral equations,, Disc. Cont. Dyn. Sys., 2005 (): 164. [3] W. Chen and C. Li, Regularity of Solutions for a system of integral equations,, Comm. Pure and Appl. Anal., 4 (2005), 1. [4] W. Chen and C. Li, The best constant in some weighted Hardy-Littlewood-Sobolev inequality,, Proc. AMS, 136 (2008), 955. [5] W. Chen and C. Li, Classification of positive solutions for nonlinear differential and integral systems with critical exponents,, Acta Mathematica Scientia, 4 (2009), 949. doi: 10.1016/S0252-9602(09)60079-5. [6] W. Chen and C. Li, An integral system and the Lane-Emden conjecture,, Disc. Cont. Dyn. Sys., 4 (2009), 1167. [7] 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. [8] W. Chen and C. Li, Super polyharmonic property of solutions for PDE systems and its applications, , submitted to Trans. AMS, (2011). [9] W. Chen and C. Li, "Methods on Nonlinear Elliptic Equations,", AIMS Book Series on Diff. Equa. & Dyn. Sys., 4 (2010). [10] C. Ma, W. Chen, and C. Li, Regularity of solutions for an integral system of Wolff type,, Advances of Math, 3 (2011), 2676. doi: 10.1016/j.aim.2010.07.020. [11] W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation,, Comm. Pure Appl. Math., LLVIII (2005), 1. [12] W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation,, Disc. Cont. Dyn. Sys., 12 (2005), 347. [13] W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations,, Comm. PDEs, 30 (2005), 59. doi: 10.1081/PDE-200044445. [14] B. Gidas, W. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n,$, Advances in Mathematics, 7a (1981). [15] F. Hang, On the integral systems related to Hardy-Littlewood-Sobolev inequality,, Math Res Lett, (2007). [16] F. Hang, X. Wang and X. Yan, An integral equation in conformal geometry,, Ann. H. Poincare Nonl. Anal., 26 (2009), 1. doi: 10.1016/j.anihpc.2007.03.006. [17] G. Hardy and J. Littelwood, Some properties of fractional integrals I,, Math. Zeitschr., 27 (1928), 565. doi: 10.1007/BF01171116. [18] C. Jin and C. Li, Symmetry of solutions to some systems of integral equations,, Proc. AMS, 134 (2006), 1661. [19] T. Kilpelaiinen and J. Maly, The Wiener test and potential estimates for quasilinear elliptic equations,, Acta Math., 172 (1994), 137. doi: 10.1007/BF02392793. [20] C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations,, Invent. Math., 123 (1996), 221. [21] E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities,, Annals of Math, 118 (1983), 349. doi: 10.2307/2007032. [22] S. Liu, Regularity, symmetry, and uniqueness of some integral type quasilinear equations,, Nonlinear Analysis: Theory, 71 (2009), 1796. [23] Y. Y. Li, Remarks on some conformally invariant integral equations: The method of moving spheres,, J. Euro. Math. Soc., 6 (2004), 153. doi: 10.4171/JEMS/6. [24] 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. [25] C. Li and L. Ma, Uniqueness of positive bound states to Shrödinger systems with critical exponents,, SIAM J. of Appl. Anal., 40 (2008), 1049. doi: 10.1137/080712301. [26] C. Liu and S. Qiao, Symmetry and monotonicity for a system of integral equations,, Comm. Pure Appl. Anal., 6 (2009), 1925. doi: 10.3934/cpaa.2009.8.1925. [27] D. Li and R. Zhuo, An integral equation on half space,, Proc. AMS, 138 (2010), 2779. [28] L. Ma and D. Chen, A Liouville type theorem for an integral system,, Comm. Pure Appl. Anal., 5 (2006), 855. doi: 10.3934/cpaa.2006.5.855. [29] L. Ma and D. Chen, Radial symmetry and monotonicity for an integral equation,, J. Math. Anal. Appl., 2 (2008), 943. doi: 10.1016/j.jmaa.2007.12.064. [30] L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation,, Arch. Rat. Mech. Anal., 2 (2010), 455. doi: 10.1007/s00205-008-0208-3. [31] N. Phuc and I. Verbitsky, Quasilinear and Hessian equations of Lane-Emden type,, Annals of Math., 168 (2008), 859. doi: 10.4007/annals.2008.168.859. [32] S. Sobolev, On a theorem of functional analysis,, Mat. Sb. (N.S.), 4 (1938), 471. [33] N. Trudinger and X. Wang, Hessian measure II,, Annals of Math., 150 (1999), 579. doi: 10.2307/121089. [34] J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations,, Math. Ann., 313 (1999), 207. doi: 10.1007/s002080050258.

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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,, C. P. A. M., XLII (1989), 271. [2] W. Chen, C. Jin, C. Li and C. Lim, Weighted Hardy-Littlewood-Sobolev inequalities and system of integral equations,, Disc. Cont. Dyn. Sys., 2005 (): 164. [3] W. Chen and C. Li, Regularity of Solutions for a system of integral equations,, Comm. Pure and Appl. Anal., 4 (2005), 1. [4] W. Chen and C. Li, The best constant in some weighted Hardy-Littlewood-Sobolev inequality,, Proc. AMS, 136 (2008), 955. [5] W. Chen and C. Li, Classification of positive solutions for nonlinear differential and integral systems with critical exponents,, Acta Mathematica Scientia, 4 (2009), 949. doi: 10.1016/S0252-9602(09)60079-5. [6] W. Chen and C. Li, An integral system and the Lane-Emden conjecture,, Disc. Cont. Dyn. Sys., 4 (2009), 1167. [7] 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. [8] W. Chen and C. Li, Super polyharmonic property of solutions for PDE systems and its applications, , submitted to Trans. AMS, (2011). [9] W. Chen and C. Li, "Methods on Nonlinear Elliptic Equations,", AIMS Book Series on Diff. Equa. & Dyn. Sys., 4 (2010). [10] C. Ma, W. Chen, and C. Li, Regularity of solutions for an integral system of Wolff type,, Advances of Math, 3 (2011), 2676. doi: 10.1016/j.aim.2010.07.020. [11] W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation,, Comm. Pure Appl. Math., LLVIII (2005), 1. [12] W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation,, Disc. Cont. Dyn. Sys., 12 (2005), 347. [13] W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations,, Comm. PDEs, 30 (2005), 59. doi: 10.1081/PDE-200044445. [14] B. Gidas, W. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n,$, Advances in Mathematics, 7a (1981). [15] F. Hang, On the integral systems related to Hardy-Littlewood-Sobolev inequality,, Math Res Lett, (2007). [16] F. Hang, X. Wang and X. Yan, An integral equation in conformal geometry,, Ann. H. Poincare Nonl. Anal., 26 (2009), 1. doi: 10.1016/j.anihpc.2007.03.006. [17] G. Hardy and J. Littelwood, Some properties of fractional integrals I,, Math. Zeitschr., 27 (1928), 565. doi: 10.1007/BF01171116. [18] C. Jin and C. Li, Symmetry of solutions to some systems of integral equations,, Proc. AMS, 134 (2006), 1661. [19] T. Kilpelaiinen and J. Maly, The Wiener test and potential estimates for quasilinear elliptic equations,, Acta Math., 172 (1994), 137. doi: 10.1007/BF02392793. [20] C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations,, Invent. Math., 123 (1996), 221. [21] E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities,, Annals of Math, 118 (1983), 349. doi: 10.2307/2007032. [22] S. Liu, Regularity, symmetry, and uniqueness of some integral type quasilinear equations,, Nonlinear Analysis: Theory, 71 (2009), 1796. [23] Y. Y. Li, Remarks on some conformally invariant integral equations: The method of moving spheres,, J. Euro. Math. Soc., 6 (2004), 153. doi: 10.4171/JEMS/6. [24] 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. [25] C. Li and L. Ma, Uniqueness of positive bound states to Shrödinger systems with critical exponents,, SIAM J. of Appl. Anal., 40 (2008), 1049. doi: 10.1137/080712301. [26] C. Liu and S. Qiao, Symmetry and monotonicity for a system of integral equations,, Comm. Pure Appl. Anal., 6 (2009), 1925. doi: 10.3934/cpaa.2009.8.1925. [27] D. Li and R. Zhuo, An integral equation on half space,, Proc. AMS, 138 (2010), 2779. [28] L. Ma and D. Chen, A Liouville type theorem for an integral system,, Comm. Pure Appl. Anal., 5 (2006), 855. doi: 10.3934/cpaa.2006.5.855. [29] L. Ma and D. Chen, Radial symmetry and monotonicity for an integral equation,, J. Math. Anal. Appl., 2 (2008), 943. doi: 10.1016/j.jmaa.2007.12.064. [30] L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation,, Arch. Rat. Mech. Anal., 2 (2010), 455. doi: 10.1007/s00205-008-0208-3. [31] N. Phuc and I. Verbitsky, Quasilinear and Hessian equations of Lane-Emden type,, Annals of Math., 168 (2008), 859. doi: 10.4007/annals.2008.168.859. [32] S. Sobolev, On a theorem of functional analysis,, Mat. Sb. (N.S.), 4 (1938), 471. [33] N. Trudinger and X. Wang, Hessian measure II,, Annals of Math., 150 (1999), 579. doi: 10.2307/121089. [34] J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations,, Math. Ann., 313 (1999), 207. doi: 10.1007/s002080050258.
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