Positive solutions of Kirchhoff type problems with critical growth on exterior domains

In this paper, we study the existence of positive solutions for a class of Kirchhoff equation with critical growth -a+b∫Ω|∇u|2dxΔu+V(x)u=u5inΩ,u∈D01,2(Ω),where a>0, b>0, V∈L32(Ω)is a given nonnegative function and Ω⊆R3is an exterior domain, that is, an unbounded domain with smooth boundary ∂Ω≠∅such that R3\Ωnon-empty and bounded. By using barycentric functions and Brouwer degree theory to prove that there exists a positive solution u∈D01,2(Ω)if R3\Ωis contained in a small ball.