Positive radial solutions for the problem with Minkowski-curvature operator on an exterior domain

We are concerned with the problem with Minkowski-curvature operator on an exterior domain

$$\begin{array}{r}\{\begin{array}{ll}-\text{div}(\frac{\mathrm{\nabla }u}{\sqrt{1-|\mathrm{\nabla }u{|}^2}})=\lambda K(|x|)\frac{f(u)}{u^{\gamma }}& \text{in}{B}^c,\\ \frac{\mathrm{\partial }u}{\mathrm{\partial }n}{|}_{\mathrm{\partial }{B}^c}=0,\underset{|x|\to \mathrm{\infty }}{lim}u(x)=0,\end{array}\end{array}(P)$$

where $0\le \gamma <1$, $B^c=\{x\in {\mathbb{R}}^N:|x|>R\}$ is a exterior domain in ${\mathbb{R}}^N$, $N>2$, $R>0$, $K\in C([R,\mathrm{\infty }),(0,\mathrm{\infty }))$ is such that ${\int }_R^{\mathrm{\infty }}rK(r)dr<\mathrm{\infty }$, the function $f:[0,\mathrm{\infty })\to (0,\mathrm{\infty })$ is a continuous function such that $\underset{s\to \mathrm{\infty }}{lim}\frac{f(s)}{s^{\gamma +1}}=0$ and $\lambda >0$ is a parameter. We show that problem $(P)$ has at least one positive radial solution for all $\lambda >0$. The proof of our main result is based upon the method of sub and super solutions.