# American Institute of Mathematical Sciences

2014, 13(2): 773-788. doi: 10.3934/cpaa.2014.13.773

## Positive solutions to involving Wolff potentials

 1 School of Mathematical Sciences, Shandong Normal University, Jinan 250014 2 School of Mathematical Sciences, Jiangsu Normal University, Xuzhou, 221116

Received  April 2013 Revised  July 2013 Published  October 2013

In this paper, we consider the weighted integral system involving Wolff potentials in $R^{n}$: \begin{eqnarray} u(x) = R_1(x)W_{\beta, \gamma}(\frac{u^pv^q(y)}{|y|^\sigma})(x), \\ v(x) = R_2(x)W_{\beta,\gamma}(\frac{v^pu^q(y)}{|y|^\sigma})(x). \end{eqnarray} where $0< R(x) \leq C$, $1 < \gamma \leq 2$, $0\leq \sigma < \beta \gamma$, $n-\beta\gamma > \sigma(\gamma-1)$, $\gamma^{*}-1=\frac{n\gamma}{n-\beta\gamma+\sigma}-1\geq 1$. Due to the weight $\frac{1}{|y|^\sigma}$, we need more complicated analytical techniques to handle the properties of the solutions. First, we use the method of regularity lifting to obtain the integrability for the solutions of this Wolff type integral equation. Next, we use the modifying and refining method of moving planes established by Chen and Li to prove the radial symmetry for the positive solutions of related integral equation. Based on these results, we obtain the decay rates of the solutions of (0.1) with $R_1(x)\equiv R_2(x)\equiv 1$ near infinity. We generalize the results in the related references.
Citation: Huan-Zhen Chen, Zhongxue Lü. Positive solutions to involving Wolff potentials. Communications on Pure & Applied Analysis, 2014, 13 (2) : 773-788. doi: 10.3934/cpaa.2014.13.773
##### 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. doi: 10.1023/A:1014845728367. [2] W. Chen and C. Li, Radial symmetry of solutions for some integral systems of Wolff type,, Disc. Cont. Dyn. Sys., 30 (2011), 1083. doi: 10.3934/dcds.2011.30.1083. [3] W. Chen and C. Li, Regularity of solutions for a system of intgral equations,, Commun. Pure Appl. Anal., 4 (2005), 1. [4] 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. [5] W. Chen and C. Li, An integral system and the Lane-Emden conjecture,, Disc. Cont. Dyn. Sys., 4 (2009), 1167. doi: 10.3934/dcds.2009.24.1167. [6] L. I. Hedberg and T. Wolff, Thin sets in nonlinear potential theory,, Ann. Inst. Fourier (Grenoble), 33 (1983), 161. [7] X. Huang, G. Hong and D. Li, Some symmetry results for integral equations involving Wolff potentials on bounded domains,, Nonlinear Anal., 75 (2012), 5601. doi: 10.1016/j.na.2012.05.007. [8] X. Huang, D. Li and L. Wang, Symmetry and monotonicity of integral equation systems,, Nonlinear Anal., 12 (2011), 3515. doi: 10.1016/j.nonrwa.2011.06.012. [9] X. Huang, D. Li and L. Wang, Radial symmetry results for systems of integral equations on $\Omega\in R^n$,, Manuscripta Math., 137 (2012), 317. doi: 10.1007/s00229-011-0465-6. [10] 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. [11] 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. [12] T. Kilpelaiinen and J. Maly, The Wiener test and potential estimates for quasilinear elliptic equations,, Acta Math., 172 (1994), 137. doi: 10.1007/BF02392793. [13] T. Kilpelaiinen and J. Maly, Degenerate elliptic equations with measure data and nonlinear potentials,, Ann. Sc. Norm. Super. Pisa Cl. Sci., 19 (1992), 591. [14] D. Labutin, Potential eatimates for a class of fully nonlinear elliptic equations,, Duke Math. J., 111 (2002), 1. doi: 10.1215/S0012-7094-02-11111-9. [15] Y. Lei, Decay rates for solutions of an integral system of Wolff type,, Potential Anal., 35 (2011), 387. 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. doi: 10.3934/dcds.2011.30.547. [17] Y. Lei and C. Li, Integrability and asymptotics of positive solutions of a $\gamma$-Laplace system,, J. Differential Equations, 252 (2012), 2739. doi: 10.1016/j.jde.2011.10.009. [18] Y. Lei and C. Ma, Asymptotic behavior for solutions of some integral equations,, Commun. Pure Appl. Anal., 10 (2011), 193. doi: 10.3934/cpaa.2011.10.193. [19] C. Li and L. Ma, Uniqueness of positive bound states to Shrödinger systems with critical exponents,, SIAM J. Math. Anal., 40 (2008), 1049. doi: 10.1137/080712301. [20] S. Liu, Regularity, symmetry, and uniqueness of some integral type quasilinear equations,, Nonlinear Anal.: Theory, 71 (2009), 1796. doi: 10.1016/j.na.2009.01.014. [21] C. Ma, W. Chen and C. Li, Regularity of solutions for an integral system of Wolff type,, Advances in Mathematics, 226 (2011), 2676. doi: 10.1016/j.aim.2010.07.020. [22] J. Maly, Wolff potential estimates of superminnimizers of Orilicz type Dirichlet integrals,, Manuscripta Math., 110 (2003), 513. doi: 10.1007/s00229-003-0358-4. [23] N. Pfuc and I. Verbitsky, Quasilinear and Hessian equations of Lane-Emden type,, Ann. of Math., 168 (2008), 859. doi: 10.4007/annals.2008.168.859. [24] E. M. Stein and G. Weiss, Fractional integrals in $n$-dimensional Euclidean space,, J. Math. Mech., 7 (1958), 503. [25] Y. Zhao and Y. Lei, Asymptotic behavior of positive solutions of a nonlinear integral system, , Nonlinear Anal., 75 (2012), 1989. doi: 10.1016/j.na.2011.09.051.

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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. doi: 10.1023/A:1014845728367. [2] W. Chen and C. Li, Radial symmetry of solutions for some integral systems of Wolff type,, Disc. Cont. Dyn. Sys., 30 (2011), 1083. doi: 10.3934/dcds.2011.30.1083. [3] W. Chen and C. Li, Regularity of solutions for a system of intgral equations,, Commun. Pure Appl. Anal., 4 (2005), 1. [4] 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. [5] W. Chen and C. Li, An integral system and the Lane-Emden conjecture,, Disc. Cont. Dyn. Sys., 4 (2009), 1167. doi: 10.3934/dcds.2009.24.1167. [6] L. I. Hedberg and T. Wolff, Thin sets in nonlinear potential theory,, Ann. Inst. Fourier (Grenoble), 33 (1983), 161. [7] X. Huang, G. Hong and D. Li, Some symmetry results for integral equations involving Wolff potentials on bounded domains,, Nonlinear Anal., 75 (2012), 5601. doi: 10.1016/j.na.2012.05.007. [8] X. Huang, D. Li and L. Wang, Symmetry and monotonicity of integral equation systems,, Nonlinear Anal., 12 (2011), 3515. doi: 10.1016/j.nonrwa.2011.06.012. [9] X. Huang, D. Li and L. Wang, Radial symmetry results for systems of integral equations on $\Omega\in R^n$,, Manuscripta Math., 137 (2012), 317. doi: 10.1007/s00229-011-0465-6. [10] 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. [11] 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. [12] T. Kilpelaiinen and J. Maly, The Wiener test and potential estimates for quasilinear elliptic equations,, Acta Math., 172 (1994), 137. doi: 10.1007/BF02392793. [13] T. Kilpelaiinen and J. Maly, Degenerate elliptic equations with measure data and nonlinear potentials,, Ann. Sc. Norm. Super. Pisa Cl. Sci., 19 (1992), 591. [14] D. Labutin, Potential eatimates for a class of fully nonlinear elliptic equations,, Duke Math. J., 111 (2002), 1. doi: 10.1215/S0012-7094-02-11111-9. [15] Y. Lei, Decay rates for solutions of an integral system of Wolff type,, Potential Anal., 35 (2011), 387. 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. doi: 10.3934/dcds.2011.30.547. [17] Y. Lei and C. Li, Integrability and asymptotics of positive solutions of a $\gamma$-Laplace system,, J. Differential Equations, 252 (2012), 2739. doi: 10.1016/j.jde.2011.10.009. [18] Y. Lei and C. Ma, Asymptotic behavior for solutions of some integral equations,, Commun. Pure Appl. Anal., 10 (2011), 193. doi: 10.3934/cpaa.2011.10.193. [19] C. Li and L. Ma, Uniqueness of positive bound states to Shrödinger systems with critical exponents,, SIAM J. Math. Anal., 40 (2008), 1049. doi: 10.1137/080712301. [20] S. Liu, Regularity, symmetry, and uniqueness of some integral type quasilinear equations,, Nonlinear Anal.: Theory, 71 (2009), 1796. doi: 10.1016/j.na.2009.01.014. [21] C. Ma, W. Chen and C. Li, Regularity of solutions for an integral system of Wolff type,, Advances in Mathematics, 226 (2011), 2676. doi: 10.1016/j.aim.2010.07.020. [22] J. Maly, Wolff potential estimates of superminnimizers of Orilicz type Dirichlet integrals,, Manuscripta Math., 110 (2003), 513. doi: 10.1007/s00229-003-0358-4. [23] N. Pfuc and I. Verbitsky, Quasilinear and Hessian equations of Lane-Emden type,, Ann. of Math., 168 (2008), 859. doi: 10.4007/annals.2008.168.859. [24] E. M. Stein and G. Weiss, Fractional integrals in $n$-dimensional Euclidean space,, J. Math. Mech., 7 (1958), 503. [25] Y. Zhao and Y. Lei, Asymptotic behavior of positive solutions of a nonlinear integral system, , Nonlinear Anal., 75 (2012), 1989. doi: 10.1016/j.na.2011.09.051.
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