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

Previous Article
Long time dynamics for forced and weakly damped KdV on the torus
CPAA Home
This Issue
Next Article
Elliptic equations involving linear and superlinear terms and critical Caffarelli-Kohn-Nirenberg exponent with sign-changing weight functions

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
Related Papers

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 & 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., (2010). Google Scholar
[2]	W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation,, Comm Pure Appl Math, 59 (2006), 330. doi: 10.1002/cpa.20116. Google Scholar
[3]	X. Chen and J. Yang, Regularity and symmetry of positive solutions of an integral system,, Acta Math. Sci., 32B (2012), 1759. Google Scholar
[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. doi: 10.1103/PhysRevLett.86.5043. Google Scholar
[5]	M. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $\mathbbR^n$,, Arch. Ration. Mech. Anal., 105 (1989), 243. doi: 10.1007/BF00251502. Google Scholar
[6]	Y. Li, Remark on some conformlly invariant integral equations: The method of moving spheres,, J. Eur. Math. Soc., 6 (2004), 153. doi: 10.4171/JEMS/6. Google Scholar
[7]	C. Li and L. Ma, Uniqueness of positive bound states to Schrödinger systems with critical exponents,, SIAM J. Math. Anal., 40 (2008), 1049. Google Scholar
[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. doi: 10.1007/s00220-005-1313-x. Google Scholar
[9]	T. C. Lin and J. Wei, Spikes in two coupled nonlinear Schrödinger equations,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 22 (2005), 403. doi: 10.1016/j.anihpc.2004.03.004. Google Scholar
[10]	L. Ma and D. Chen, Radial symmetry and monotonicity for an integral equation,, J. Math. Anal. Appl., 342 (2008), 943. doi: 10.1016/j.jmaa.2007.12.064. Google Scholar
[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. doi: 10.1016/j.mcm.2008.06.010. Google Scholar
[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. doi: 10.1016/j.jde.2008.04008. Google Scholar
[13]	Y. Zhao and Y. Lei, Asymptotic behavior of positive solutions of a nonlinear integral system,, Nonlinear Anal., 75 (2012), 1989. doi: 10.1016/j.na.2011.09.051. Google Scholar

show all references

References:

[1]	W. Chen and C. Li, "Methods on Nolinear Elliptic Equation,", AIMS Ser. Differ. Dyn. Syst., (2010). Google Scholar
[2]	W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation,, Comm Pure Appl Math, 59 (2006), 330. doi: 10.1002/cpa.20116. Google Scholar
[3]	X. Chen and J. Yang, Regularity and symmetry of positive solutions of an integral system,, Acta Math. Sci., 32B (2012), 1759. Google Scholar
[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. doi: 10.1103/PhysRevLett.86.5043. Google Scholar
[5]	M. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $\mathbbR^n$,, Arch. Ration. Mech. Anal., 105 (1989), 243. doi: 10.1007/BF00251502. Google Scholar
[6]	Y. Li, Remark on some conformlly invariant integral equations: The method of moving spheres,, J. Eur. Math. Soc., 6 (2004), 153. doi: 10.4171/JEMS/6. Google Scholar
[7]	C. Li and L. Ma, Uniqueness of positive bound states to Schrödinger systems with critical exponents,, SIAM J. Math. Anal., 40 (2008), 1049. Google Scholar
[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. doi: 10.1007/s00220-005-1313-x. Google Scholar
[9]	T. C. Lin and J. Wei, Spikes in two coupled nonlinear Schrödinger equations,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 22 (2005), 403. doi: 10.1016/j.anihpc.2004.03.004. Google Scholar
[10]	L. Ma and D. Chen, Radial symmetry and monotonicity for an integral equation,, J. Math. Anal. Appl., 342 (2008), 943. doi: 10.1016/j.jmaa.2007.12.064. Google Scholar
[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. doi: 10.1016/j.mcm.2008.06.010. Google Scholar
[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. doi: 10.1016/j.jde.2008.04008. Google Scholar
[13]	Y. Zhao and Y. Lei, Asymptotic behavior of positive solutions of a nonlinear integral system,, Nonlinear Anal., 75 (2012), 1989. doi: 10.1016/j.na.2011.09.051. Google Scholar

[1]	Julián López-Gómez. Uniqueness of radially symmetric large solutions. Conference Publications, 2007, 2007 (Special) : 677-686. doi: 10.3934/proc.2007.2007.677
[2]	Xiaolong Han, Guozhen Lu. Regularity of solutions to an integral equation associated with Bessel potential. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1111-1119. doi: 10.3934/cpaa.2011.10.1111
[3]	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
[4]	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
[5]	Xavier Cabré. Elliptic PDE's in probability and geometry: Symmetry and regularity of solutions. Discrete & Continuous Dynamical Systems - A, 2008, 20 (3) : 425-457. doi: 10.3934/dcds.2008.20.425
[6]	M.T. Boudjelkha. Extended Riemann Bessel functions. Conference Publications, 2005, 2005 (Special) : 121-130. doi: 10.3934/proc.2005.2005.121
[7]	Gui-Qiang Chen, Bo Su. Discontinuous solutions for Hamilton-Jacobi equations: Uniqueness and regularity. Discrete & Continuous Dynamical Systems - A, 2003, 9 (1) : 167-192. doi: 10.3934/dcds.2003.9.167
[8]	Alberto Fiorenza, Anna Mercaldo, Jean Michel Rakotoson. Regularity and uniqueness results in grand Sobolev spaces for parabolic equations with measure data. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 893-906. doi: 10.3934/dcds.2002.8.893
[9]	Florin Catrina, Zhi-Qiang Wang. Asymptotic uniqueness and exact symmetry of k-bump solutions for a class of degenerate elliptic problems. Conference Publications, 2001, 2001 (Special) : 80-87. doi: 10.3934/proc.2001.2001.80
[10]	Mason A. Porter, Richard L. Liboff. The radially vibrating spherical quantum billiard. Conference Publications, 2001, 2001 (Special) : 310-318. doi: 10.3934/proc.2001.2001.310
[11]	Kim-Ngan Le, William McLean, Martin Stynes. Existence, uniqueness and regularity of the solution of the time-fractional Fokker–Planck equation with general forcing. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2765-2787. doi: 10.3934/cpaa.2019124
[12]	Yutian Lei. Positive solutions of integral systems involving Bessel potentials. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2721-2737. doi: 10.3934/cpaa.2013.12.2721
[13]	Jesse Goodman, Daniel Spector. Some remarks on boundary operators of Bessel extensions. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 493-509. doi: 10.3934/dcdss.2018027
[14]	Yonggang Zhao, Mingxin Wang. An integral equation involving Bessel potentials on half space. Communications on Pure & Applied Analysis, 2015, 14 (2) : 527-548. doi: 10.3934/cpaa.2015.14.527
[15]	Mingchun Wang, Jiankai Xu, Huoxiong Wu. On Positive solutions of integral equations with the weighted Bessel potentials. Communications on Pure & Applied Analysis, 2019, 18 (2) : 625-641. doi: 10.3934/cpaa.2019031
[16]	Thomas I. Vogel. Comments on radially symmetric liquid bridges with inflected profiles. Conference Publications, 2005, 2005 (Special) : 862-867. doi: 10.3934/proc.2005.2005.862
[17]	Jun Xie, Jinlong Yuan, Dongxia Wang, Weili Liu, Chongyang Liu. Uniqueness of solutions to fuzzy relational equations regarding Max-av composition and strong regularity of the matrices in Max-av algebra. Journal of Industrial & Management Optimization, 2018, 14 (3) : 1007-1022. doi: 10.3934/jimo.2017087
[18]	Lu Chen, Zhao Liu, Guozhen Lu. Qualitative properties of solutions to an integral system associated with the Bessel potential. Communications on Pure & Applied Analysis, 2016, 15 (3) : 893-906. doi: 10.3934/cpaa.2016.15.893
[19]	Ali Akgül, Mustafa Inc, Esra Karatas. Reproducing kernel functions for difference equations. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1055-1064. doi: 10.3934/dcdss.2015.8.1055
[20]	Chiara Corsato, Colette De Coster, Pierpaolo Omari. Radially symmetric solutions of an anisotropic mean curvature equation modeling the corneal shape. Conference Publications, 2015, 2015 (special) : 297-303. doi: 10.3934/proc.2015.0297

2018 Impact Factor: 0.925

American Institute of Mathematical Sciences