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 \(\begin{aligned} \left\{ \begin{aligned}&-\left( a+b \int _{\Omega }|\nabla u|^{2} d x\right) \Delta u+V(x) u=u^{5}&\text{ in } \Omega , \\&u\in D^{1,2}_0(\Omega ), \end{aligned}\right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{ccc}& -\left(a,+,b,{\int }_{\mathrm{\Omega }},{|\mathrm{\nabla }u|}^2,d,x\right)\mathrm{\Delta }u+V(x)u=u^5\hfill & \hfill \text{in}\mathrm{\Omega },\\ \hfill & u\in D_0^{1,2}(\mathrm{\Omega }),\hfill \end{array}\right)\end{array}$$ where \(a>0\) $a>0$ , \(b>0\) $b>0$ , \(V\in L^\frac{3}{2}(\Omega )\) $V\in L^{\frac{3}{2}}(\mathrm{\Omega })$ is a given nonnegative function and \(\Omega \subseteq \mathbb {R}^3\) $\mathrm{\Omega }\subseteq {\mathbb{R}}^3$ is an exterior domain, that is, an unbounded domain with smooth boundary \(\partial \Omega \ne \emptyset \) $\partial \mathrm{\Omega }\ne \mathrm{\varnothing }$ such that \(\mathbb {R}^3\backslash \Omega \) ${\mathbb{R}}^3\backslash \mathrm{\Omega }$ non-empty and bounded. By using barycentric functions and Brouwer degree theory to prove that there exists a positive solution \(u\in D^{1,2}_0(\Omega )\) $u\in D_0^{1,2}(\mathrm{\Omega })$ if \(\mathbb {R}^3\backslash \Omega \) ${\mathbb{R}}^3\backslash \mathrm{\Omega }$ is contained in a small ball.