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November  2013, 12(6): 2685-2696. doi: 10.3934/cpaa.2013.12.2685

## Regularity, symmetry and uniqueness of positive solutions to a nonlinear elliptic system

 1 Department of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi 330022, China, China, China

Received  September 2012 Revised  December 2012 Published  May 2013

In this paper, we are concerned with properties of positive solutions of the following fractional elliptic system \begin{eqnarray} {(-\Delta+I)}^{\frac{\alpha}{2}}u=\frac{u^pv^q}{|x|^\beta}, \quad {(-\Delta+I)}^{\frac{\alpha}{2}}v=\frac{v^pu^q}{|x|^\beta}\quad in\quad R^n, \end{eqnarray} where $n \geq 3$, $0 \le \beta < \alpha < n$, $p, q>1$ and $p+q<\frac{n+\alpha-\beta}{n-\alpha+\beta}$. We show that positive solutions of the system are radially symmetric and belong to $L^\infty(R^n)$, which possibly implies that the solutions are locally Hölder continuous. Moreover, if $\alpha=2, \beta =0,p\le q$, we show that positive solution pair $(u,v)$ of the system is unique and $u=v = U$, where $U$ is the unique positive solution of the problem \begin{eqnarray} -\Delta u + u = u^{p+q}\quad {\rm in}\quad \mathbb{R}^n. \end{eqnarray}
Citation: Shiren Zhu, Xiaoli Chen, Jianfu Yang. Regularity, symmetry and uniqueness of positive solutions to a nonlinear elliptic system. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2685-2696. doi: 10.3934/cpaa.2013.12.2685
##### References:
 [1] W. Chen and C. Li, "Methods on Nolinear Elliptic Equation," AIMS Ser. Differ. Dyn. Syst., vol.4, AIMS, 2010. [2] 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. [3] X. Chen and J. Yang, Regularity and symmetry of positive solutions of an integral system, Acta Math. Sci., 32B (2012), 1759-1780. [4] T. Kanna and M. Lakshmanan, Exact soliton solutions, shape changing collisions, and partially coherent solitons in coupled nonlinear Schrödinger equations, Phys. Rev. Lett., 86 (2001), 5043-5046. doi: 10.1103/PhysRevLett.86.5043. [5] M. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $\mathbbR^n$, Arch. Ration. Mech. Anal., 105 (1989), 243-266. doi: 10.1007/BF00251502. [6] Y. Li, Remark on some conformlly invariant integral equations: The method of moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180. doi: 10.4171/JEMS/6. [7] C. Li and L. Ma, Uniqueness of positive bound states to Schrödinger systems with critical exponents, SIAM J. Math. Anal., 40 (2008), 1049-1057. [8] T. C. Lin and J. Wei, Ground state of N coupled nonlinear Schrödinger equations in $\mathbbR^n, n\leq 3$, Commun. Math. Phys., 255 (2005), 629-653. doi: 10.1007/s00220-005-1313-x. [9] T. C. Lin and J. Wei, Spikes in two coupled nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 403-439. doi: 10.1016/j.anihpc.2004.03.004. [10] L. Ma and D. Chen, Radial symmetry and monotonicity for an integral equation, J. Math. Anal. Appl., 342 (2008), 943-949. doi: 10.1016/j.jmaa.2007.12.064. [11] L. Ma and D. Chen, Radial symmetry and uniqueness for positive solutions of a Schrödinger type systems, Mathematical and Computer Modelling, 49 (2009), 379-385. doi: 10.1016/j.mcm.2008.06.010. [12] L. Ma and L. Zhao, Uniqueness of ground states of some coupled nonlinear Schrödinger systems and their application, J. Diffe. Equa., 245 (2008), 2551-2565. doi: 10.1016/j.jde.2008.04008. [13] Y. Zhao and Y. Lei, Asymptotic behavior of positive solutions of a nonlinear integral system, Nonlinear Anal., 75 (2012), 1989-1999. doi: 10.1016/j.na.2011.09.051.

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##### References:
 [1] W. Chen and C. Li, "Methods on Nolinear Elliptic Equation," AIMS Ser. Differ. Dyn. Syst., vol.4, AIMS, 2010. [2] 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. [3] X. Chen and J. Yang, Regularity and symmetry of positive solutions of an integral system, Acta Math. Sci., 32B (2012), 1759-1780. [4] T. Kanna and M. Lakshmanan, Exact soliton solutions, shape changing collisions, and partially coherent solitons in coupled nonlinear Schrödinger equations, Phys. Rev. Lett., 86 (2001), 5043-5046. doi: 10.1103/PhysRevLett.86.5043. [5] M. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $\mathbbR^n$, Arch. Ration. Mech. Anal., 105 (1989), 243-266. doi: 10.1007/BF00251502. [6] Y. Li, Remark on some conformlly invariant integral equations: The method of moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180. doi: 10.4171/JEMS/6. [7] C. Li and L. Ma, Uniqueness of positive bound states to Schrödinger systems with critical exponents, SIAM J. Math. Anal., 40 (2008), 1049-1057. [8] T. C. Lin and J. Wei, Ground state of N coupled nonlinear Schrödinger equations in $\mathbbR^n, n\leq 3$, Commun. Math. Phys., 255 (2005), 629-653. doi: 10.1007/s00220-005-1313-x. [9] T. C. Lin and J. Wei, Spikes in two coupled nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 403-439. doi: 10.1016/j.anihpc.2004.03.004. [10] L. Ma and D. Chen, Radial symmetry and monotonicity for an integral equation, J. Math. Anal. Appl., 342 (2008), 943-949. doi: 10.1016/j.jmaa.2007.12.064. [11] L. Ma and D. Chen, Radial symmetry and uniqueness for positive solutions of a Schrödinger type systems, Mathematical and Computer Modelling, 49 (2009), 379-385. doi: 10.1016/j.mcm.2008.06.010. [12] L. Ma and L. Zhao, Uniqueness of ground states of some coupled nonlinear Schrödinger systems and their application, J. Diffe. Equa., 245 (2008), 2551-2565. doi: 10.1016/j.jde.2008.04008. [13] Y. Zhao and Y. Lei, Asymptotic behavior of positive solutions of a nonlinear integral system, Nonlinear Anal., 75 (2012), 1989-1999. doi: 10.1016/j.na.2011.09.051.
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