Multiple positive solutions for a Kirchhoff type equation involving two potentials

We consider the existence of positive solutions to the following Kirchhoff type equation with critical exponent −a+b∫ℝ3|∇u|2dxΔu=λV(x)u+K(x)u5,u∈D1,2(ℝ3), where a>0, b>0, and λ>0 is a parameter. Under some assumptions, two positive solutions are obtained by means of variational methods. It is worth emphasizing that for the local minimal solution, we use an entirely new method, and its some properties are also obtained.