In this paper, we discuss the existence and uniqueness of radial solutions of the elliptic equation with nonlinear gradient term \( \left \{ \textstyle\begin{array}{l} -\triangle u = f\big(|x|,\;u,\;\frac{x}{|x|}\cdot \nabla u\big), \qquad x\in \Omega , \\ u|_{\partial \Omega}=0\,, \end{array}\displaystyle \right . \) where $\Omega =\{x\in \mathbb{R}^{N}:\;r_{1}<|x|<r_{2}\}$ is an annular domain in $\mathbb{R}^{N}$ , $N\ge 3$ , $f:[r_{1},\,r_{2}]\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ is continuous. Under an inequality condition of $f(r,\,\xi ,\,\eta )$ and a Nagumo-type growth of $f(r,\,\xi ,\,\eta )$ on η, an existence result of radial solutions is obtained. The inequality condition is related to the principal eigenvalue $\lambda _{1}$ of the Laplace operator −△ under the boundary condition $u|_{\partial \Omega}=0$ , is optimal and allow that the nonlinearity $f(r,\,\xi ,\,\eta )$ is downwards superlinear growth on ξ and η, and the Nagumo-type growth restricts $f(r,\,\xi ,\,\eta )$ is at most quadratic growth on η. When the inequality condition is properly strengthened, the uniqueness result is obtained.
