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

Abstract

In this paper, we study the existence of positive solutions for a class of Kirchhoff equation with critical growth

$$\begin{aligned} \left\{ \begin{aligned}&-\left( a+b \int _{\Omega }|\nabla u|^{2} d x\right) \Delta u+V(x) u=u^{5}&\text{ in } \Omega , \\&u\in D^{1,2}_0(\Omega ), \end{aligned}\right. \end{aligned}$$

where \(a>0\), \(b>0\), \(V\in L^\frac{3}{2}(\Omega )\) is a given nonnegative function and \(\Omega \subseteq \mathbb {R}^3\) is an exterior domain, that is, an unbounded domain with smooth boundary \(\partial \Omega \ne \emptyset \) such that \(\mathbb {R}^3\backslash \Omega \) non-empty and bounded. By using barycentric functions and Brouwer degree theory to prove that there exists a positive solution \(u\in D^{1,2}_0(\Omega )\) if \(\mathbb {R}^3\backslash \Omega \) is contained in a small ball.