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 $ -(a+b \sum_{\Omega_e} | \nabla u |^p dx) \Delta_p u = \lambda f (|x|, u), x ∈ \Omega_e, u=0 $ on $ \delta \Omega_e $, where λ > 0 is a parameter, $ Ω_e = {x ∈ \mathbb{R}^N : |x| > r_0}, r_0 > 0, N > p > 1, Δp $ is the p-Laplacian operator, and $ f ∈ C([r_0,+∞) × [0,+∞) , \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 λ.