# American Institute of Mathematical Sciences

March  2016, 15(2): 385-398. doi: 10.3934/cpaa.2016.15.385

## Existence and nonexistence of positive solutions to an integral system involving Wolff potential

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

Received  March 2015 Revised  October 2015 Published  January 2016

In this paper, we are concerned with the sufficient and necessary conditions for the existence and nonexistence of the positive solutions of the following system involving Wolff type potential: \begin{eqnarray} & u(x) =c_{1}(x)W_{\beta,\gamma}(v^{q})(x), \\ &v(x) =c_{2}(x)W_{\alpha,\tau}(u^{p})(x). \end{eqnarray} Here $x\in R^{n}$, $1 < \gamma,\tau \leq 2$, $\alpha,\beta > 0$, $0< \beta\gamma$, $\alpha \tau < n$, and the functions $c_{1}(x),c_{2}(x)$ are double bounded. This system is helpful to well understand some nonlinear PDEs and other nonlinear problems. Different from the case of $\alpha=\beta,\gamma=\tau$, it is more difficult to handle the critical condition. Fortunately, by applying the special iteration scheme and some critical asymptotic analysis, we establish the sharp criteria for existence and nonexistence of positive solutions to system (0.1). Then, we use the method of moving planes to prove the symmetry and monotonicity for the positive solutions of (0.1) when $c_{1}(x)\equiv c_{2}(x)\equiv1$ in the case \begin{eqnarray} \frac{\gamma-1}{p+\gamma-1}+\frac{\tau-1}{q+\tau-1}=\frac{n-\alpha\tau}{2n-\alpha\tau+\beta\gamma} +\frac{n-\beta\gamma}{2n-\beta\gamma+\alpha\tau}. \end{eqnarray}
Citation: Wu Chen, Zhongxue Lu. Existence and nonexistence of positive solutions to an integral system involving Wolff potential. Communications on Pure & Applied Analysis, 2016, 15 (2) : 385-398. doi: 10.3934/cpaa.2016.15.385
##### References:
 [1] C. Cascante, J. Ortega and I. Verbitsky, Wolff's inequality for radially nonincreasing kernels and applications to trace inequalities,, \emph{Potential Analysis.}, 16 (2002), 347.  doi: 10.1023/A:1014845728367.  Google Scholar [2] L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth,, \emph{Comm. Pure Appl. Math.}, 42 (1989), 271.  doi: 10.1002/cpa.3160420304.  Google Scholar [3] W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation,, \emph{Comm. Pure Appl. Math.}, 59 (2006), 330.  doi: 10.1002/cpa.20116.  Google Scholar [4] W. Chen and C. Li, Regularity of solutions for a system of integral equations,, \emph{Comm. Pure Appl. Anal.}, 4 (2005), 1.   Google Scholar [5] W. Chen and C. Li, Radial symmetry of solutions for some integral systems of Wolff type,, \emph{Disc. Cont. Dyn. Sys.}, 30 (2011), 1083.  doi: 10.3934/dcds.2011.30.1083.  Google Scholar [6] W. Chen and C. Li, Super polyharmonic property of solutions for PDE systems and its applications,, \emph{Comm. Pure Appl. Anal.}, 12 (2013), 2497.  doi: 10.3934/cpaa.2013.12.2497.  Google Scholar [7] L. I. Hedberg and T. Wolff, Thin sets in nonlinear potential theory,, \emph{Ann. Inst. Fourier (Grenoble)}, 33 (1983), 161.   Google Scholar [8] X. Huang, D. Li and L. Wang, Existence and symmetry of positive solutions of an integral equation system,, \emph{Math. Comput. Modelling}, 52 (2010), 892.  doi: 10.1016/j.mcm.2010.05.020.  Google Scholar [9] X. Huang, G. Hong and D. Li, Some symmetry results for integral equations involving Wolff potential on bounded domains,, \emph{Nonlinear Anal.}, 75 (2012), 5601.  doi: 10.1016/j.na.2012.05.007.  Google Scholar [10] T. Kilpelaiinen and J. Maly, The Wiener test and potential estimates for quasilinear ellipitc equations,, \emph{Acta Math.}, 172 (1994), 137.  doi: 10.1007/BF02392793.  Google Scholar [11] T. Kilpelaiinen and J. Maly, Degenerate elliptic equations with measure data and nonlinear potentials,, \emph{Ann. Sc. Norm. Super. Pisa Cl. Sci.}, 19 (1992), 591.   Google Scholar [12] D. Labutin, Potential eatimates for a class of fully nonlinear elliptic equations,, \emph{Duke Math. J.}, 111 (2002), 1.  doi: 10.1215/S0012-7094-02-11111-9.  Google Scholar [13] Y. Lei, C. Li and C. Ma, Decay estimation for positve solutions of a $\gamma$-Laplace equation,, \emph{Disc. Cont. Dyn. Syst.}, 30 (2011), 547.  doi: 10.3934/dcds.2011.30.547.  Google Scholar [14] Y. Lei and C. Li, Sharp criteria of Liouville type for some nonlinear systems,, arXiv:1301.6235., ().   Google Scholar [15] Y. Lei, Decay rates for solutions of an integral system of Wolff type,, \emph{Potential Analysis}, 35 (2011), 387.  doi: 10.1007/s11118-010-9218-5.  Google Scholar [16] C. Ma, W. Chen and C. Li, Regularity of solutions for an integral system of Wolff type,, \emph{Advances in Mathematics}, 226 (2011), 2676.  doi: 10.1016/j.aim.2010.07.020.  Google Scholar [17] N. Phuc and I. Verbitsky, Quasilinear and Hessian equations of Lane-Emden type,, \emph{Ann. of Math.}, 168 (2008), 859.  doi: 10.4007/annals.2008.168.859.  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 inequalities,, \emph{Potential Analysis.}, 16 (2002), 347.  doi: 10.1023/A:1014845728367.  Google Scholar [2] L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth,, \emph{Comm. Pure Appl. Math.}, 42 (1989), 271.  doi: 10.1002/cpa.3160420304.  Google Scholar [3] W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation,, \emph{Comm. Pure Appl. Math.}, 59 (2006), 330.  doi: 10.1002/cpa.20116.  Google Scholar [4] W. Chen and C. Li, Regularity of solutions for a system of integral equations,, \emph{Comm. Pure Appl. Anal.}, 4 (2005), 1.   Google Scholar [5] W. Chen and C. Li, Radial symmetry of solutions for some integral systems of Wolff type,, \emph{Disc. Cont. Dyn. Sys.}, 30 (2011), 1083.  doi: 10.3934/dcds.2011.30.1083.  Google Scholar [6] W. Chen and C. Li, Super polyharmonic property of solutions for PDE systems and its applications,, \emph{Comm. Pure Appl. Anal.}, 12 (2013), 2497.  doi: 10.3934/cpaa.2013.12.2497.  Google Scholar [7] L. I. Hedberg and T. Wolff, Thin sets in nonlinear potential theory,, \emph{Ann. Inst. Fourier (Grenoble)}, 33 (1983), 161.   Google Scholar [8] X. Huang, D. Li and L. Wang, Existence and symmetry of positive solutions of an integral equation system,, \emph{Math. Comput. Modelling}, 52 (2010), 892.  doi: 10.1016/j.mcm.2010.05.020.  Google Scholar [9] X. Huang, G. Hong and D. Li, Some symmetry results for integral equations involving Wolff potential on bounded domains,, \emph{Nonlinear Anal.}, 75 (2012), 5601.  doi: 10.1016/j.na.2012.05.007.  Google Scholar [10] T. Kilpelaiinen and J. Maly, The Wiener test and potential estimates for quasilinear ellipitc equations,, \emph{Acta Math.}, 172 (1994), 137.  doi: 10.1007/BF02392793.  Google Scholar [11] T. Kilpelaiinen and J. Maly, Degenerate elliptic equations with measure data and nonlinear potentials,, \emph{Ann. Sc. Norm. Super. Pisa Cl. Sci.}, 19 (1992), 591.   Google Scholar [12] D. Labutin, Potential eatimates for a class of fully nonlinear elliptic equations,, \emph{Duke Math. J.}, 111 (2002), 1.  doi: 10.1215/S0012-7094-02-11111-9.  Google Scholar [13] Y. Lei, C. Li and C. Ma, Decay estimation for positve solutions of a $\gamma$-Laplace equation,, \emph{Disc. Cont. Dyn. Syst.}, 30 (2011), 547.  doi: 10.3934/dcds.2011.30.547.  Google Scholar [14] Y. Lei and C. Li, Sharp criteria of Liouville type for some nonlinear systems,, arXiv:1301.6235., ().   Google Scholar [15] Y. Lei, Decay rates for solutions of an integral system of Wolff type,, \emph{Potential Analysis}, 35 (2011), 387.  doi: 10.1007/s11118-010-9218-5.  Google Scholar [16] C. Ma, W. Chen and C. Li, Regularity of solutions for an integral system of Wolff type,, \emph{Advances in Mathematics}, 226 (2011), 2676.  doi: 10.1016/j.aim.2010.07.020.  Google Scholar [17] N. Phuc and I. Verbitsky, Quasilinear and Hessian equations of Lane-Emden type,, \emph{Ann. of Math.}, 168 (2008), 859.  doi: 10.4007/annals.2008.168.859.  Google Scholar
 [1] 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 [2] Meixia Dou. A direct method of moving planes for fractional Laplacian equations in the unit ball. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1797-1807. doi: 10.3934/cpaa.2016015 [3] Baiyu Liu. Direct method of moving planes for logarithmic Laplacian system in bounded domains. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5339-5349. doi: 10.3934/dcds.2018235 [4] Pengyan Wang, Pengcheng Niu. A direct method of moving planes for a fully nonlinear nonlocal system. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1707-1718. doi: 10.3934/cpaa.2017082 [5] Haiyan Wang. Existence and nonexistence of positive radial solutions for quasilinear systems. Conference Publications, 2009, 2009 (Special) : 810-817. doi: 10.3934/proc.2009.2009.810 [6] Antonio Greco, Vincenzino Mascia. Non-local sublinear problems: Existence, comparison, and radial symmetry. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 503-519. doi: 10.3934/dcds.2019021 [7] Xiaotao Huang, Lihe Wang. Radial symmetry results for Bessel potential integral equations in exterior domains and in annular domains. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1121-1134. doi: 10.3934/cpaa.2017054 [8] Miaomiao Cai, Li Ma. Moving planes for nonlinear fractional Laplacian equation with negative powers. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4603-4615. doi: 10.3934/dcds.2018201 [9] Yutian Lei. Wolff type potential estimates and application to nonlinear equations with negative exponents. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2067-2078. doi: 10.3934/dcds.2015.35.2067 [10] 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 [11] Pei Ma, Yan Li, Jihui Zhang. Symmetry and nonexistence of positive solutions for fractional systems. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1053-1070. doi: 10.3934/cpaa.2018051 [12] Xiaoxue Ji, Pengcheng Niu, Pengyan Wang. Non-existence results for cooperative semi-linear fractional system via direct method of moving spheres. Communications on Pure & Applied Analysis, 2020, 19 (2) : 1111-1128. doi: 10.3934/cpaa.2020051 [13] Orlando Lopes. Uniqueness and radial symmetry of minimizers for a nonlocal variational problem. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2265-2282. doi: 10.3934/cpaa.2019102 [14] Ran Zhuo, Yan Li. Nonexistence and symmetry of solutions for Schrödinger systems involving fractional Laplacian. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1595-1611. doi: 10.3934/dcds.2019071 [15] Leyun Wu, Pengcheng Niu. Symmetry and nonexistence of positive solutions to fractional p-Laplacian equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1573-1583. doi: 10.3934/dcds.2019069 [16] Dario D. Monticelli, Fabio Punzo. Nonexistence results for elliptic differential inequalities with a potential in bounded domains. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 675-695. doi: 10.3934/dcds.2018029 [17] Patricio Felmer, César Torres. Radial symmetry of ground states for a regional fractional Nonlinear Schrödinger Equation. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2395-2406. doi: 10.3934/cpaa.2014.13.2395 [18] Sara Barile, Addolorata Salvatore. Radial solutions of semilinear elliptic equations with broken symmetry on unbounded domains. Conference Publications, 2013, 2013 (special) : 41-49. doi: 10.3934/proc.2013.2013.41 [19] Dongbing Zha. Remarks on nonlinear elastic waves in the radial symmetry in 2-D. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 4051-4062. doi: 10.3934/dcds.2016.36.4051 [20] Jian Hao, Zhilin Li, Sharon R. Lubkin. An augmented immersed interface method for moving structures with mass. Discrete & Continuous Dynamical Systems - B, 2012, 17 (4) : 1175-1184. doi: 10.3934/dcdsb.2012.17.1175

2018 Impact Factor: 0.925