American Institute of Mathematical Sciences

Advanced
2005, 2005(Special): 510-517. doi: 10.3934/proc.2005.2005.510

On lane-emden type systems

Philip Korman 1, and  Junping Shi 2,

1. 

Department Of Mathematical Sciences, University Of Cincinnati, Cincinnati Ohio 45221-0025
2. 

Department of Mathematics, College of William and Mary, Williamsburg, Virginia 23187, United States

Received  September 2004 Revised  April 2005 Published  September 2005

We consider a class of singular systems of Lane-Emden type \begin{equation} \nonumber \begin{cases} \Delta u + \la u^{p_1} v^{q_1}=0, & x\in D,\\ \Delta v + \la u^{p_2} v^{q_2}=0, & x\in D,\\ u=v=0, & x\in \partial D, \end{cases} \end{equation} with $p_1\le 0, \; p_2> 0, \; q_1> 0, \; q_2\le 0$, and $D$ a smooth domain in $\R^n$. In case the system is sublinear we prove existence of a positive solution. If $D$ is a ball in $\R^n$, we prove both existence and uniqueness of positive radially symmetric solution.
Keywords: semilinear elliptic system, uniqueness..
Mathematics Subject Classification: Primary: 35J55, 35J4.
Citation: Philip Korman, Junping Shi. On lane-emden type systems. Conference Publications, 2005, 2005 (Special) : 510-517. doi: 10.3934/proc.2005.2005.510
[1]

Rafael Ortega, James R. Ward Jr. A semilinear elliptic system with vanishing nonlinearities. Conference Publications, 2003, 2003 (Special) : 688-693. doi: 10.3934/proc.2003.2003.688
[2]

Yimei Li, Jiguang Bao. Semilinear elliptic system with boundary singularity. Discrete & Continuous Dynamical Systems - A, 2020, 40 (4) : 2189-2212. doi: 10.3934/dcds.2020111
[3]

Ruofei Yao, Yi Li, Hongbin Chen. Uniqueness of positive radial solutions of a semilinear elliptic equation in an annulus. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1585-1594. doi: 10.3934/dcds.2018122
[4]

Dongho Chae. Existence of a semilinear elliptic system with exponential nonlinearities. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 709-718. doi: 10.3934/dcds.2007.18.709
[5]

Diane Denny. A unique positive solution to a system of semilinear elliptic equations. Conference Publications, 2013, 2013 (special) : 193-195. doi: 10.3934/proc.2013.2013.193
[6]

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
[7]

Piero Montecchiari, Paul H. Rabinowitz. A nondegeneracy condition for a semilinear elliptic system and the existence of 1- bump solutions. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 6995-7012. doi: 10.3934/dcds.2019241
[8]

Sami Baraket, Soumaya Sâanouni, Nihed Trabelsi. Singular limit solutions for a 2-dimensional semilinear elliptic system of Liouville type in some general case. Discrete & Continuous Dynamical Systems - A, 2020, 40 (2) : 1013-1063. doi: 10.3934/dcds.2020069
[9]

Philip Korman. On uniqueness of positive solutions for a class of semilinear equations. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 865-871. doi: 10.3934/dcds.2002.8.865
[10]

Carmen Cortázar, Marta García-Huidobro, Pilar Herreros. On the uniqueness of bound state solutions of a semilinear equation with weights. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6761-6784. doi: 10.3934/dcds.2019294
[11]

Lei Wei, Zhaosheng Feng. Isolated singularity for semilinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 3239-3252. doi: 10.3934/dcds.2015.35.3239
[12]

Maria Francesca Betta, Olivier Guibé, Anna Mercaldo. Uniqueness for Neumann problems for nonlinear elliptic equations. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1023-1048. doi: 10.3934/cpaa.2019050
[13]

Miao Chen, Youyan Wan, Chang-Lin Xiang. Local uniqueness problem for a nonlinear elliptic equation. Communications on Pure & Applied Analysis, 2020, 19 (2) : 1037-1055. doi: 10.3934/cpaa.2020048
[14]

C. Cortázar, Marta García-Huidobro. On the uniqueness of ground state solutions of a semilinear equation containing a weighted Laplacian. Communications on Pure & Applied Analysis, 2006, 5 (4) : 813-826. doi: 10.3934/cpaa.2006.5.813
[15]

C. Cortázar, Marta García-Huidobro. On the uniqueness of ground state solutions of a semilinear equation containing a weighted Laplacian. Communications on Pure & Applied Analysis, 2006, 5 (1) : 71-84. doi: 10.3934/cpaa.2006.5.71
[16]

Tomás Caraballo, María J. Garrido–Atienza, Björn Schmalfuss, José Valero. Asymptotic behaviour of a stochastic semilinear dissipative functional equation without uniqueness of solutions. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 439-455. doi: 10.3934/dcdsb.2010.14.439
[17]

Irena Lasiecka, Roberto Triggiani. Global exact controllability of semilinear wave equations by a double compactness/uniqueness argument. Conference Publications, 2005, 2005 (Special) : 556-565. doi: 10.3934/proc.2005.2005.556
[18]

Jaan Janno, Kairi Kasemets. Uniqueness for an inverse problem for a semilinear time-fractional diffusion equation. Inverse Problems & Imaging, 2017, 11 (1) : 125-149. doi: 10.3934/ipi.2017007
[19]

Junping Shi, R. Shivaji. Semilinear elliptic equations with generalized cubic nonlinearities. Conference Publications, 2005, 2005 (Special) : 798-805. doi: 10.3934/proc.2005.2005.798
[20]

Xavier Cabré, Manel Sanchón, Joel Spruck. A priori estimates for semistable solutions of semilinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 601-609. doi: 10.3934/dcds.2016.36.601

 Impact Factor: 

Tools

Metrics

  • PDF downloads (13)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences