We are concerned with the problem with Minkowski-curvature operator on an exterior domain \( \left\{ \begin{array}{ll} -\textrm{div}\Big (\phi _{N}(\nabla u(x))\Big )=\lambda a(|x|)f(u(x)),~~~\textrm{in}~B^{c},\\ \frac{\partial u}{\partial n}\mid _{\partial B^{c}}=0,~~\lim \limits _{|x|\rightarrow \infty }u(x)=0, \end{array} \right. \) where \(\frac{\partial u}{\partial n}\) denotes the outward normal derivative of u, \(\phi _{N}(y)=\frac{y}{\sqrt{1-|y|^{2}}},~y\in \mathbb {R}^{N},~B^{c}=\{x\in \mathbb {R}^{N}:|x|>R\}\) is an exterior domain in \(\mathbb {R}^{N},~N\ge 3,~R>0,~a\in L^{1}_{loc}((0,1],\mathbb {R})\) may change sign, \(\lambda \) is a nonnegative real parameter, \(f\in C([0,\infty ),\mathbb {R})\) with \(f(0)>0\) . The proof of the main result is based on the Leray-Schauder fixed point theorem.
