Positive solutions for weakly coupled nonlinear elliptic systems

In this article, we consider the existence of positive solutions for weakly coupled nonlinear elliptic systems equation(0.1){-Δu+u=(1+a(x))|u|p-1u+μ|u|α-2u|v|β+λv    in ℝN,-Δv+v=(1+b(x))|v|p-1v+μ|u|α|v|β-2v+λu​​​​​   in ℝN,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.