A new observation on the positive solutions for Kirchhoff equations in the exterior of a ball

We consider the existence of positive solutions for following Kirchhoff equation $\left(\begin{array}{cc}-\left(a+b{\int }_{\Omega }{|\nabla u|}^{2}dx\right)\Delta u+u={\left|u\right|}^{p-2}u& \text{in}\Omega ,\\ u=0& \text{on}\partial \Omega ,\end{array}\right)$ where $a,b>0$, $\Omega =\{x\in R^N:\left|x\right|>1\}$ is the exterior of the unit ball in $R^N$ and $N\ge 2$. It is well-known that if $4<p<\infty$, by standard minimization method on the Nehari manifold, one can obtain a positive radial solution. In present paper, we prove the existence of positive radial solutions for $2<p\le 4$. This is the first contribution to the Kirchhoff equation in exterior domains provided that $2<p\le 4.$