数学物理学报(英文版) ›› 2010, Vol. 30 ›› Issue (5): 1577-1592.doi: 10.1016/S0252-9602(10)60151-8

• 论文 • 上一篇    下一篇

POSITIVE SOLUTIONS FOR WEAKLY COUPLED NONLINEAR ELLIPTIC SYSTEMS

龙静, 杨健夫   

  1. Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, |Wuhan 430071, China
    Graduate School, Chinese Academy of Sciences, Beijing 100049, China; Department of Mathematics, Jiangxi Normal University, Nanchang 330022, China
  • 出版日期:2010-09-20 发布日期:2010-09-20
  • 基金资助:

    This work is supported by National Natural Sciences Foundations of China (10631030, 10961016).

POSITIVE SOLUTIONS FOR WEAKLY COUPLED NONLINEAR ELLIPTIC SYSTEMS

 LONG Jing, YANG Jian-Fu   

  • Online:2010-09-20 Published:2010-09-20
  • Supported by:

    This work is supported by National Natural Sciences Foundations of China (10631030, 10961016).

摘要:

In this article, we consider the existence of positive solutions for weakly coupled nonlinear elliptic systems
$$\left\{ \begin{array}{ll}-\Delta u+u=(1+a(x))|u|^{p-1}u+\mu|u|^{\alpha-2}u|v|^\beta+\lambda v
\quad & \rm{in}\ {\Bbb R}^N,\\
-\Delta v+v=(1+b(x))|v|^{p-1}v+\mu|u|^\alpha|v|^{\beta-2}v+\lambda u
& \rm{in} \ {\Bbb R}^N. \end{array} \right.\eqno(0.1)$$
To find nontrivial solutions, we first investigate autonomous systems. In this case, results of bifurcation from semi-trivial solutions are obtained by the implicit function theorem. Next, the existence of positive solutions of problem (0.1) is obtained by variational methods.

关键词: Elliptic system in RN,  positive solution, bifurcation, variational method

Abstract:

In this article, we consider the existence of positive solutions for weakly coupled nonlinear elliptic systems
$$\left\{ \begin{array}{ll}-\Delta u+u=(1+a(x))|u|^{p-1}u+\mu|u|^{\alpha-2}u|v|^\beta+\lambda v
\quad & \rm{in}\ {\Bbb R}^N,\\
-\Delta v+v=(1+b(x))|v|^{p-1}v+\mu|u|^\alpha|v|^{\beta-2}v+\lambda u
& \rm{in} \ {\Bbb R}^N. \end{array} \right.\eqno(0.1)$$
To find nontrivial solutions, we first investigate autonomous systems. In this case, results of bifurcation from semi-trivial solutions are obtained by the implicit function theorem. Next, the existence of positive solutions of problem (0.1) is obtained by variational methods.

Key words: Elliptic system in RN,  positive solution, bifurcation, variational method

中图分类号: 

  • 35J50