# American Institute of Mathematical Sciences

May  2014, 34(5): 1879-1904. doi: 10.3934/dcds.2014.34.1879

## The properties of positive solutions to an integral system involving Wolff potential

 1 School of Mathematical Sciences, Jiangsu Normal University, Xuzhou, 221116, China

Received  January 2013 Revised  June 2013 Published  October 2013

In this paper, we consider the positive solutions of the following weighted integral system involving Wolff potential in $R^{n}$: $$\left\{ \begin{array}{ll} u(x) = R_1(x)W_{\beta,\gamma}(\frac{v^q}{|y|^{\sigma}})(x), \\ v(x) = R_2(x)W_{\beta,\gamma}(\frac{u^p}{|y|^{\sigma}})(x). (0.1) \end{array} \right.$$ This system is helpful to understand some nonlinear PDEs and other nonlinear problems. Different from the case of $\sigma=0$, it is difficult to handle the properties of the solutions since there is singularity at origin. First, we overcome this difficulty by modifying and refining the new method which was introduced to explore the integrability result establishes by Ma, Chen and Li, and obtain an optimal integrability. Second, we use the method of moving planes to prove the radial symmetry for the positive solutions of (0.1) when $R_{1}(x)\equiv R_{2}(x)\equiv 1$. Based on these results, by intricate analytical techniques, we obtain the estimate of the decay rates of those solutions near infinity.
Citation: Huan Chen, Zhongxue Lü. The properties of positive solutions to an integral system involving Wolff potential. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 1879-1904. doi: 10.3934/dcds.2014.34.1879
##### References:
 [1] C. Cascante, J. Ortega and I. Verbitsky, Wolff's inequality for radially nonincreasing kernels and applications to trace inequalityes, Potential Anal., 16 (2002), 347-372. doi: 10.1023/A:1014845728367. [2] W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations, Commun. Partial Differential Equations, 30 (2005), 59-65. doi: 10.1081/PDE-200044445. [3] W. Chen and C. Li, Regularity of solutions for a system of intgral equations, Commun. Pure Appl. Anal., 4 (2005), 1-8. [4] 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. [5] W. Chen and C. Li, An integral system and the Lane-Emden conjecture, Disc. Cont. Dyn. Sys., 24 (2009), 1167-1184. doi: 10.3934/dcds.2009.24.1167. [6] W. Chen and C. Li, Radial symmetry of solutions for some integral systems of Wolff type, Disc. Cont. Dyn. Sys., 30 (2011), 1083-1093. doi: 10.3934/dcds.2011.30.1083. [7] G. Hardy and J. Littelwood, Some properties of fractional integral (1), Math. Z., 27 (1928), 565-606. doi: 10.1007/BF01171116. [8] L. I. Hedberg and T. Wolff, Thin sets in nonlinear potential theory, Ann. Inst. Fourier (Grenoble), 33 (1983), 161-187. doi: 10.5802/aif.944. [9] C. Jin and C. Li, Symmetry of solutions to some syetems of integral equations, Proc. Amer. Math. Soc., 134 (2006), 1661-1670. doi: 10.1090/S0002-9939-05-08411-X. [10] C. Jin and C. Li, Qualitative analysis of some systems of integral equations, Calc. Var. PDEs, 26 (2006), 447-457. doi: 10.1007/s00526-006-0013-5. [11] T. Kilpelaiinen and J. Maly, The Wiener test and potential estimates for quasilinear ellipitc equations, Acta Math., 172 (1994), 137-161. doi: 10.1007/BF02392793. [12] T. Kilpelaiinen and J. Maly, Degenerate elliptic equations with measure data and nonlinear potentials, Ann. Sc. Norm. Super. Pisa Cl. Sci., 19 (1992), 591-613. [13] D. Labutin, Potential eatimates for a class of fully nonlinear elliptic equations, Duke Math. J., 111 (2002), 1-49. doi: 10.1215/S0012-7094-02-11111-9. [14] LY. Lei and C. Li, Integrability and asymptotics of positive solutions of a $\gamma$-Laplace system, J. Differential Equations, 252 (2012), 2739-2758. doi: 10.1016/j.jde.2011.10.009. [15] Y. Lei, Decay rates for solutions of an integral system of Wolff type, Potential Analysis, 35 (2011), 387-402. doi: 10.1007/s11118-010-9218-5. [16] Y. Lei, C. Li and C. Ma, Decay estimation for positve solutions of a $\gamma$-Laplace equation, Disc. Cont. Dyn. Sys., 30 (2011), 547-558. doi: 10.3934/dcds.2011.30.547. [17] C. Li and L. Ma, Uniqueness of positive bound states to Shrodinger systems with critical exponents, SIAM J. of Appl. Anal., 40 (2008), 1049-1057. doi: 10.1137/080712301. [18] C. Ma, W. Chen and C. Li, Regularity of solutions for an integral system of Wolff type, Advances in Mathematics, 226 (2011), 2676-2699. doi: 10.1016/j.aim.2010.07.020. [19] N. Pfuc and I. Verbitsky, Quasilinear and Hessian equations of Lane-Emden type, Ann. of Math., 168 (2008), 859-914. doi: 10.4007/annals.2008.168.859. [20] S. Sobolev, On a theorem of functional analysis, Mat. Sb. (N.S.), 4 (1938), 471-479, Amer. Math. Soc. Transl. Ser., 234 (1963), 39-68. [21] S. Sun and Y. Lei, Fast decay estimates for integrable solutions of the Lane-Emden type integral systems involving the Wolff potentials, Journal of Functional Analysis, 263 (2012), 3857-3882. doi: 10.1016/j.jfa.2012.09.012. [22] E. M. Stein and G. Weiss, Fractional integrals in $n$-dimensional Euclidean space, J. Math. Mech., 7 (1958), 503-514.

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##### References:
 [1] C. Cascante, J. Ortega and I. Verbitsky, Wolff's inequality for radially nonincreasing kernels and applications to trace inequalityes, Potential Anal., 16 (2002), 347-372. doi: 10.1023/A:1014845728367. [2] W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations, Commun. Partial Differential Equations, 30 (2005), 59-65. doi: 10.1081/PDE-200044445. [3] W. Chen and C. Li, Regularity of solutions for a system of intgral equations, Commun. Pure Appl. Anal., 4 (2005), 1-8. [4] 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. [5] W. Chen and C. Li, An integral system and the Lane-Emden conjecture, Disc. Cont. Dyn. Sys., 24 (2009), 1167-1184. doi: 10.3934/dcds.2009.24.1167. [6] W. Chen and C. Li, Radial symmetry of solutions for some integral systems of Wolff type, Disc. Cont. Dyn. Sys., 30 (2011), 1083-1093. doi: 10.3934/dcds.2011.30.1083. [7] G. Hardy and J. Littelwood, Some properties of fractional integral (1), Math. Z., 27 (1928), 565-606. doi: 10.1007/BF01171116. [8] L. I. Hedberg and T. Wolff, Thin sets in nonlinear potential theory, Ann. Inst. Fourier (Grenoble), 33 (1983), 161-187. doi: 10.5802/aif.944. [9] C. Jin and C. Li, Symmetry of solutions to some syetems of integral equations, Proc. Amer. Math. Soc., 134 (2006), 1661-1670. doi: 10.1090/S0002-9939-05-08411-X. [10] C. Jin and C. Li, Qualitative analysis of some systems of integral equations, Calc. Var. PDEs, 26 (2006), 447-457. doi: 10.1007/s00526-006-0013-5. [11] T. Kilpelaiinen and J. Maly, The Wiener test and potential estimates for quasilinear ellipitc equations, Acta Math., 172 (1994), 137-161. doi: 10.1007/BF02392793. [12] T. Kilpelaiinen and J. Maly, Degenerate elliptic equations with measure data and nonlinear potentials, Ann. Sc. Norm. Super. Pisa Cl. Sci., 19 (1992), 591-613. [13] D. Labutin, Potential eatimates for a class of fully nonlinear elliptic equations, Duke Math. J., 111 (2002), 1-49. doi: 10.1215/S0012-7094-02-11111-9. [14] LY. Lei and C. Li, Integrability and asymptotics of positive solutions of a $\gamma$-Laplace system, J. Differential Equations, 252 (2012), 2739-2758. doi: 10.1016/j.jde.2011.10.009. [15] Y. Lei, Decay rates for solutions of an integral system of Wolff type, Potential Analysis, 35 (2011), 387-402. doi: 10.1007/s11118-010-9218-5. [16] Y. Lei, C. Li and C. Ma, Decay estimation for positve solutions of a $\gamma$-Laplace equation, Disc. Cont. Dyn. Sys., 30 (2011), 547-558. doi: 10.3934/dcds.2011.30.547. [17] C. Li and L. Ma, Uniqueness of positive bound states to Shrodinger systems with critical exponents, SIAM J. of Appl. Anal., 40 (2008), 1049-1057. doi: 10.1137/080712301. [18] C. Ma, W. Chen and C. Li, Regularity of solutions for an integral system of Wolff type, Advances in Mathematics, 226 (2011), 2676-2699. doi: 10.1016/j.aim.2010.07.020. [19] N. Pfuc and I. Verbitsky, Quasilinear and Hessian equations of Lane-Emden type, Ann. of Math., 168 (2008), 859-914. doi: 10.4007/annals.2008.168.859. [20] S. Sobolev, On a theorem of functional analysis, Mat. Sb. (N.S.), 4 (1938), 471-479, Amer. Math. Soc. Transl. Ser., 234 (1963), 39-68. [21] S. Sun and Y. Lei, Fast decay estimates for integrable solutions of the Lane-Emden type integral systems involving the Wolff potentials, Journal of Functional Analysis, 263 (2012), 3857-3882. doi: 10.1016/j.jfa.2012.09.012. [22] E. M. Stein and G. Weiss, Fractional integrals in $n$-dimensional Euclidean space, J. Math. Mech., 7 (1958), 503-514.
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