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

Abstract

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 $\lambda$.