Multiple positive solutions to a Kirchhoff type problem involving a critical nonlinearity

In this paper, we investigate a class of Kirchhoff type problem involving a critical nonlinearity −(1+b∫Ω|∇u|2dx)△u=λu+|u|4u,u∈H01(Ω), where b>0, λ>λ1, λ1 is the principal eigenvalue of (−△,H01(Ω)). With the help of the Nehari manifold, we obtain the multiplicity of positive solutions for λ in a small right neighborhood of λ1 and prove that one of the solutions is a positive ground state solution, which is different from the result of Brézis-Nirenberg in 1983. This paper can be regarded as the complementary work of Naimen (2015).