Acta mathematica scientia,Series B ›› 2010, Vol. 30 ›› Issue (5): 1577-1592.doi: 10.1016/S0252-9602(10)60151-8

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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).

Abstract:

In this article, we consider the existence of positive solutions for weakly coupled nonlinear elliptic systems
{Δu+u=(1+a(x))|u|p1u+μ|u|α2u|v|β+λvin RN,Δv+v=(1+b(x))|v|p1v+μ|u|α|v|β2v+λuin RN.\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

CLC Number: 

  • 35J50
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