This paper deals with the existence of positive radial solutions of the elliptic equation with nonlinear gradient term \(\begin{aligned} \left\{ \begin{array}{ll} -\triangle u = f(|x|,\;u,\;|\nabla u|),\qquad x\in \Omega ,\\ u|_{\partial \Omega }=0\,, \end{array}\right. \end{aligned}\) where \(\Omega =\{x\in \mathbb {R}^N:\;r_1<|x|<r_2\}\) , \(N\ge 3\) , \(f:[r_1,\,r_2]\times \mathbb {R}^+\times \mathbb {R}\rightarrow \mathbb {R}^+\) is continuous. Under some inequality conditions of \(f(x,\,\xi ,\,\eta )\) when \(|(\xi ,\,\eta )|\) is small and large, two existence results of positive radial solutions are obtained. These inequality conditions are related to the principal eigenvalue \(\lambda _1\) of the corresponding linear eigenvalue problem, are optimal and allow that the nonlinearity \(f(r,\,\xi ,\,\eta )\) is superlinear or sublinear growth on \(\xi \) and \(\eta \) as \(|(\xi ,\,\eta )|\rightarrow 0\) or \(|(\xi ,\,\eta )|\rightarrow \infty \) . The main results are proved by the fixed point index theory in cones.
