Existence and uniqueness of radial solutions for elliptic equations with nonlinear gradient terms

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.