Existence of positive radial solutions to a p-Laplacian Kirchhoff type problem on the exterior of a ball

In this paper the authors study the existence of positive radial solutions to the Kirchhoff type problem involving the \(p\)-Laplacian \[-\Big(a+b\int_{\Omega_e}|\nabla u|^p dx\Big)\Delta_p u=\lambda f\left(|x|,u\right),\ x\in \Omega_e,\quad u=0\ \text{on} \ \partial\Omega_e,\] where \(\lambda \gt 0\) is a parameter, \(\Omega_e = \lbrace x\in\mathbb{R}^N : |x|\gt r_0\rbrace\), \(r_0\gt 0\), \(N \gt p \gt 1\), \(\Delta_p\) is the \(p\)-Laplacian operator, and \(f\in C(\left[ r_0, +\infty\right)\times\left[0,+\infty\right),\mathbb{R})\) is a non-decreasing function with respect to its second variable. By using the Mountain Pass Theorem, they prove the existence of positive radial solutions for small values of \(\lambda\).