Existence and nonexistence of positive radial solutions for a class of $p$-Laplacian superlinear problems with nonlinear boundary conditions

We prove the existence of positive radial solutions to the problem \begin{document}$ \begin{cases} -\Delta _{p}u = \lambda \ K(|x|)f(u)\ \text{in } |x|>r_{0}, \\ \dfrac{\partial u}{\partial n}+\tilde{c}(u)u = 0\ \text{on }|x| = r_{0},\ \ u(x)\rightarrow 0\text{ as }|x|\rightarrow \infty ,\end{cases} $\end{document} where \begin{document}$ \ \Delta _{p}u = div(|\nabla u|^{p-2}\nabla u),\ N>p>1, \Omega = \{x\in \mathbb{R}^{N}:|x|>r_{0}>0\}, $\end{document} \begin{document}$ f:(0,\infty )\rightarrow \mathbb{R} $\end{document} is \begin{document}$ p $\end{document} -superlinear at \begin{document}$ \infty $\end{document} with possible singularity at \begin{document}$ 0, $\end{document} and \begin{document}$ \lambda $\end{document} is a small positive parameter. A nonexistence result is also established when \begin{document}$ f $\end{document} has semipositone structure at \begin{document}$ 0. $\end{document}