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

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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

Shiren Zhu ^1, , Xiaoli Chen ^1, and Jianfu Yang ^1,

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

Received September 2012 Revised December 2012 Published May 2013

Abstract
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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}

Keywords: uniqueness, Bessel kernel., Regularity, radially symmetry.

Mathematics Subject Classification: Primary: 35J25, 47G30, 35B453, 35J7.

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.

show all references

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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