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 & Continuous Dynamical Systems - A, 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.  doi: 10.1023/A:1014845728367.  Google Scholar

[2]

W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations,, Commun. Partial Differential Equations, 30 (2005), 59.  doi: 10.1081/PDE-200044445.  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., 24 (2009), 1167.  doi: 10.3934/dcds.2009.24.1167.  Google Scholar

[6]

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

[7]

G. Hardy and J. Littelwood, Some properties of fractional integral (1),, Math. Z., 27 (1928), 565.  doi: 10.1007/BF01171116.  Google Scholar

[8]

L. I. Hedberg and T. Wolff, Thin sets in nonlinear potential theory,, Ann. Inst. Fourier (Grenoble), 33 (1983), 161.  doi: 10.5802/aif.944.  Google Scholar

[9]

C. Jin and C. Li, Symmetry of solutions to some syetems of integral equations,, Proc. Amer. Math. Soc., 134 (2006), 1661.  doi: 10.1090/S0002-9939-05-08411-X.  Google Scholar

[10]

C. Jin and C. Li, Qualitative analysis of some systems of integral equations,, Calc. Var. PDEs, 26 (2006), 447.  doi: 10.1007/s00526-006-0013-5.  Google Scholar

[11]

T. Kilpelaiinen and J. Maly, The Wiener test and potential estimates for quasilinear ellipitc equations,, Acta Math., 172 (1994), 137.  doi: 10.1007/BF02392793.  Google Scholar

[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.   Google Scholar

[13]

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

[14]

LY. 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

[15]

Y. Lei, Decay rates for solutions of an integral system of Wolff type,, Potential Analysis, 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]

C. Li and L. Ma, Uniqueness of positive bound states to Shrodinger systems with critical exponents,, SIAM J. of Appl. Anal., 40 (2008), 1049.  doi: 10.1137/080712301.  Google Scholar

[18]

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

[19]

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

[20]

S. Sobolev, On a theorem of functional analysis,, Mat. Sb. (N.S.), 4 (1938), 471.   Google Scholar

[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.  doi: 10.1016/j.jfa.2012.09.012.  Google Scholar

[22]

E. M. Stein and G. Weiss, Fractional integrals in $n$-dimensional Euclidean space,, J. Math. Mech., 7 (1958), 503.   Google Scholar

show all references

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, C. Li and B. Ou, Classification of solutions for a system of integral equations,, Commun. Partial Differential Equations, 30 (2005), 59.  doi: 10.1081/PDE-200044445.  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., 24 (2009), 1167.  doi: 10.3934/dcds.2009.24.1167.  Google Scholar

[6]

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

[7]

G. Hardy and J. Littelwood, Some properties of fractional integral (1),, Math. Z., 27 (1928), 565.  doi: 10.1007/BF01171116.  Google Scholar

[8]

L. I. Hedberg and T. Wolff, Thin sets in nonlinear potential theory,, Ann. Inst. Fourier (Grenoble), 33 (1983), 161.  doi: 10.5802/aif.944.  Google Scholar

[9]

C. Jin and C. Li, Symmetry of solutions to some syetems of integral equations,, Proc. Amer. Math. Soc., 134 (2006), 1661.  doi: 10.1090/S0002-9939-05-08411-X.  Google Scholar

[10]

C. Jin and C. Li, Qualitative analysis of some systems of integral equations,, Calc. Var. PDEs, 26 (2006), 447.  doi: 10.1007/s00526-006-0013-5.  Google Scholar

[11]

T. Kilpelaiinen and J. Maly, The Wiener test and potential estimates for quasilinear ellipitc equations,, Acta Math., 172 (1994), 137.  doi: 10.1007/BF02392793.  Google Scholar

[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.   Google Scholar

[13]

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

[14]

LY. 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

[15]

Y. Lei, Decay rates for solutions of an integral system of Wolff type,, Potential Analysis, 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]

C. Li and L. Ma, Uniqueness of positive bound states to Shrodinger systems with critical exponents,, SIAM J. of Appl. Anal., 40 (2008), 1049.  doi: 10.1137/080712301.  Google Scholar

[18]

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

[19]

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

[20]

S. Sobolev, On a theorem of functional analysis,, Mat. Sb. (N.S.), 4 (1938), 471.   Google Scholar

[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.  doi: 10.1016/j.jfa.2012.09.012.  Google Scholar

[22]

E. M. Stein and G. Weiss, Fractional integrals in $n$-dimensional Euclidean space,, J. Math. Mech., 7 (1958), 503.   Google Scholar

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