# 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.   Google Scholar [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.   Google Scholar [3] W. Chen and C. Li, Regularity of Solutions for a system of integral equations,, Comm. Pure and Appl. Anal., 4 (2005), 1.   Google Scholar [4] W. Chen and C. Li, The best constant in some weighted Hardy-Littlewood-Sobolev inequality,, Proc. AMS, 136 (2008), 955.   Google Scholar [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.  Google Scholar [6] W. Chen and C. Li, An integral system and the Lane-Emden conjecture,, Disc. Cont. Dyn. Sys., 4 (2009), 1167.   Google Scholar [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.  Google Scholar [8] W. Chen and C. Li, Super polyharmonic property of solutions for PDE systems and its applications, , submitted to Trans. AMS, (2011).   Google Scholar [9] W. Chen and C. Li, "Methods on Nonlinear Elliptic Equations,", AIMS Book Series on Diff. Equa. & Dyn. Sys., 4 (2010).   Google Scholar [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.  Google Scholar [11] W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation,, Comm. Pure Appl. Math., LLVIII (2005), 1.   Google Scholar [12] W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation,, Disc. Cont. Dyn. Sys., 12 (2005), 347.   Google Scholar [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.  Google Scholar [14] B. Gidas, W. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n,$, Advances in Mathematics, 7a (1981).   Google Scholar [15] F. Hang, On the integral systems related to Hardy-Littlewood-Sobolev inequality,, Math Res Lett, (2007).   Google Scholar [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.  Google Scholar [17] G. Hardy and J. Littelwood, Some properties of fractional integrals I,, Math. Zeitschr., 27 (1928), 565.  doi: 10.1007/BF01171116.  Google Scholar [18] C. Jin and C. Li, Symmetry of solutions to some systems of integral equations,, Proc. AMS, 134 (2006), 1661.   Google Scholar [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.  Google Scholar [20] C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations,, Invent. Math., 123 (1996), 221.   Google Scholar [21] E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities,, Annals of Math, 118 (1983), 349.  doi: 10.2307/2007032.  Google Scholar [22] S. Liu, Regularity, symmetry, and uniqueness of some integral type quasilinear equations,, Nonlinear Analysis: Theory, 71 (2009), 1796.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [27] D. Li and R. Zhuo, An integral equation on half space,, Proc. AMS, 138 (2010), 2779.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [32] S. Sobolev, On a theorem of functional analysis,, Mat. Sb. (N.S.), 4 (1938), 471.   Google Scholar [33] N. Trudinger and X. Wang, Hessian measure II,, Annals of Math., 150 (1999), 579.  doi: 10.2307/121089.  Google Scholar [34] 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,, C. P. A. M., XLII (1989), 271.   Google Scholar [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.   Google Scholar [3] W. Chen and C. Li, Regularity of Solutions for a system of integral equations,, Comm. Pure and Appl. Anal., 4 (2005), 1.   Google Scholar [4] W. Chen and C. Li, The best constant in some weighted Hardy-Littlewood-Sobolev inequality,, Proc. AMS, 136 (2008), 955.   Google Scholar [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.  Google Scholar [6] W. Chen and C. Li, An integral system and the Lane-Emden conjecture,, Disc. Cont. Dyn. Sys., 4 (2009), 1167.   Google Scholar [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.  Google Scholar [8] W. Chen and C. Li, Super polyharmonic property of solutions for PDE systems and its applications, , submitted to Trans. AMS, (2011).   Google Scholar [9] W. Chen and C. Li, "Methods on Nonlinear Elliptic Equations,", AIMS Book Series on Diff. Equa. & Dyn. Sys., 4 (2010).   Google Scholar [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.  Google Scholar [11] W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation,, Comm. Pure Appl. Math., LLVIII (2005), 1.   Google Scholar [12] W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation,, Disc. Cont. Dyn. Sys., 12 (2005), 347.   Google Scholar [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.  Google Scholar [14] B. Gidas, W. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n,$, Advances in Mathematics, 7a (1981).   Google Scholar [15] F. Hang, On the integral systems related to Hardy-Littlewood-Sobolev inequality,, Math Res Lett, (2007).   Google Scholar [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.  Google Scholar [17] G. Hardy and J. Littelwood, Some properties of fractional integrals I,, Math. Zeitschr., 27 (1928), 565.  doi: 10.1007/BF01171116.  Google Scholar [18] C. Jin and C. Li, Symmetry of solutions to some systems of integral equations,, Proc. AMS, 134 (2006), 1661.   Google Scholar [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.  Google Scholar [20] C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations,, Invent. Math., 123 (1996), 221.   Google Scholar [21] E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities,, Annals of Math, 118 (1983), 349.  doi: 10.2307/2007032.  Google Scholar [22] S. Liu, Regularity, symmetry, and uniqueness of some integral type quasilinear equations,, Nonlinear Analysis: Theory, 71 (2009), 1796.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [27] D. Li and R. Zhuo, An integral equation on half space,, Proc. AMS, 138 (2010), 2779.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [32] S. Sobolev, On a theorem of functional analysis,, Mat. Sb. (N.S.), 4 (1938), 471.   Google Scholar [33] N. Trudinger and X. Wang, Hessian measure II,, Annals of Math., 150 (1999), 579.  doi: 10.2307/121089.  Google Scholar [34] 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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