Existence and uniqueness of radial solution for the elliptic equation system in an annulus

This article discusses the existence and uniqueness of radial solution for the elliptic equation system

$$\{\begin{array}{l}-\mathrm{△}u=f(|x|,u,v,|\mathrm{\nabla }u|),x\in \mathrm{\Omega },\\ -\mathrm{△}v=g(|x|,u,v,|\mathrm{\nabla }v|),x\in \mathrm{\Omega },\\ u{|}_{\mathrm{\partial }\mathrm{\Omega }}=0,v{|}_{\mathrm{\partial }\mathrm{\Omega }}=0,\end{array}$$

where $\mathrm{\Omega }=\{x\in {\mathbb{R}}^N:r_1<|x|<r_2\},N\ge 3,f,g:[r_1,r_2]\times \mathbb{R}\times \mathbb{R}\times {\mathbb{R}}^{+}\to \mathbb{R}$ are continuous. Due to the appearance of the gradient term in the nonlinearity, the equation system has no variational structure and the variational method cannot be applied to it directly. We will give the correlation conditions of $f$ and $g$, that is, $f$ and $g$ are superlinear or sublinear, and prove the existence and uniqueness of radial solutions by using Leray-Schauder fixed point theorem.