# American Institute of Mathematical Sciences

March  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. Google Scholar [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. Google Scholar [3] W. Chen and C. Li, Regularity of solutions for a system of intgral equations,, Commun. Pure Appl. Anal., 4 (2005), 1. Google Scholar [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. Google Scholar [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. Google Scholar [6] L. I. Hedberg and T. Wolff, Thin sets in nonlinear potential theory,, Ann. Inst. Fourier (Grenoble), 33 (1983), 161. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [24] E. M. Stein and G. Weiss, Fractional integrals in $n$-dimensional Euclidean space,, J. Math. Mech., 7 (1958), 503. Google Scholar [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. Google Scholar

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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. Google Scholar [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. Google Scholar [3] W. Chen and C. Li, Regularity of solutions for a system of intgral equations,, Commun. Pure Appl. Anal., 4 (2005), 1. Google Scholar [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. Google Scholar [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. Google Scholar [6] L. I. Hedberg and T. Wolff, Thin sets in nonlinear potential theory,, Ann. Inst. Fourier (Grenoble), 33 (1983), 161. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [24] E. M. Stein and G. Weiss, Fractional integrals in $n$-dimensional Euclidean space,, J. Math. Mech., 7 (1958), 503. Google Scholar [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. Google Scholar
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