We are concerned with the quasilinear Neumann problems in Euclidean space \(\begin{aligned} \left\{ \begin{array}{ll} -\text {div}\Big (\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\Big )=\lambda a(|x|)f(u)\ \ \ & \text {in}\ \mathcal {A},\\ \frac{\partial u}{\partial \nu }=0 \ \ & \text {on}\ \partial \mathcal {A}, \end{array} \right. \qquad \qquad \qquad {(P)} \end{aligned}\) where \(\frac{\partial u}{\partial \nu }\) denotes the outward normal derivative of u, \(\mathcal {A}=\{x\in \mathbb {R}^N\mid R_1<|x|<R_2\}\) is an annular domain in \(\mathbb {R}^N\) , \(N\ge 2\) , \(R_2>R_1>0\) , a(|x|) is a radial sign-changing function, the function f is superlinear at 0 and \(\lambda >0\) is a parameter. We show that the existence of an unbounded connected set of positive radial regular solutions \((\lambda ,u)\) of (P) bifurcating from \(u=0\) as \(\lambda \rightarrow +\infty \) . The proof of our main result is based upon the method of lower and upper solutions.
