# 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 and 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, Potential Analysis., 16 (2002), 347-372. doi: 10.1023/A:1014845728367. [2] L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math., 42 (1989), 271-297. doi: 10.1002/cpa.3160420304. [3] W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343. doi: 10.1002/cpa.20116. [4] W. Chen and C. Li, Regularity of solutions for a system of integral equations, Comm. Pure Appl. Anal., 4 (2005),1-8. [5] 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. [6] W. Chen and C. Li, Super polyharmonic property of solutions for PDE systems and its applications, Comm. Pure Appl. Anal., 12 (2013), 2497-2514. doi: 10.3934/cpaa.2013.12.2497. [7] L. I. Hedberg and T. Wolff, Thin sets in nonlinear potential theory, Ann. Inst. Fourier (Grenoble), 33 (1983), 161-187. [8] X. Huang, D. Li and L. Wang, Existence and symmetry of positive solutions of an integral equation system, Math. Comput. Modelling, 52 (2010), 892-901. doi: 10.1016/j.mcm.2010.05.020. [9] X. Huang, G. Hong and D. Li, Some symmetry results for integral equations involving Wolff potential on bounded domains, Nonlinear Anal., 75 (2012), 5601-5611. doi: 10.1016/j.na.2012.05.007. [10] 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. [11] 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. [12] 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. [13] Y. Lei, C. Li and C. Ma, Decay estimation for positve solutions of a $\gamma$-Laplace equation, Disc. Cont. Dyn. Syst., 30 (2011), 547-558. doi: 10.3934/dcds.2011.30.547. [14] Y. Lei and C. Li, Sharp criteria of Liouville type for some nonlinear systems, arXiv:1301.6235. [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] 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. [17] N. Phuc 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.

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##### References:
 [1] C. Cascante, J. Ortega and I. Verbitsky, Wolff's inequality for radially nonincreasing kernels and applications to trace inequalities, Potential Analysis., 16 (2002), 347-372. doi: 10.1023/A:1014845728367. [2] L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math., 42 (1989), 271-297. doi: 10.1002/cpa.3160420304. [3] W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343. doi: 10.1002/cpa.20116. [4] W. Chen and C. Li, Regularity of solutions for a system of integral equations, Comm. Pure Appl. Anal., 4 (2005),1-8. [5] 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. [6] W. Chen and C. Li, Super polyharmonic property of solutions for PDE systems and its applications, Comm. Pure Appl. Anal., 12 (2013), 2497-2514. doi: 10.3934/cpaa.2013.12.2497. [7] L. I. Hedberg and T. Wolff, Thin sets in nonlinear potential theory, Ann. Inst. Fourier (Grenoble), 33 (1983), 161-187. [8] X. Huang, D. Li and L. Wang, Existence and symmetry of positive solutions of an integral equation system, Math. Comput. Modelling, 52 (2010), 892-901. doi: 10.1016/j.mcm.2010.05.020. [9] X. Huang, G. Hong and D. Li, Some symmetry results for integral equations involving Wolff potential on bounded domains, Nonlinear Anal., 75 (2012), 5601-5611. doi: 10.1016/j.na.2012.05.007. [10] 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. [11] 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. [12] 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. [13] Y. Lei, C. Li and C. Ma, Decay estimation for positve solutions of a $\gamma$-Laplace equation, Disc. Cont. Dyn. Syst., 30 (2011), 547-558. doi: 10.3934/dcds.2011.30.547. [14] Y. Lei and C. Li, Sharp criteria of Liouville type for some nonlinear systems, arXiv:1301.6235. [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] 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. [17] N. Phuc 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.
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