Acta mathematica scientia, Series B >
POSITIVE SOLUTIONS FOR WEAKLY COUPLED NONLINEAR ELLIPTIC SYSTEMS
Online published: 2010-09-20
Supported by
This work is supported by National Natural Sciences Foundations of China (10631030, 10961016).
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
LONG Jing , YANG Jian-Fu . POSITIVE SOLUTIONS FOR WEAKLY COUPLED NONLINEAR ELLIPTIC SYSTEMS[J]. Acta mathematica scientia, Series B, 2010 , 30(5) : 1577 -1592 . DOI: 10.1016/S0252-9602(10)60151-8
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