Existence and multiplicity of solutions for nonlocal systems with   Kirchhoff type
  Firstly, we use Nehari manifold and Mountain Pass Lemma to prove an existence result of positive solutions for a class of nonlocal elliptic system with Kirchhoff type. Then a multiplicity result is established by cohomological index of Fadell and Rabinowitz. We also consider the critical case and prove existence of positive least energy solution when the parameter $\beta$ is sufficiently large. 
Zhitao Zhang, Yimin Sun
math.AP