The existence and uniqueness of radial solutions are discussed for elliptic boundary value problems \left\{\begin{array}{l}-\mathrm{\Delta }u=f(\left|x\right|,u,\left|?u\right|),x\in \mathrm{\Omega },\\ u\left|{}_{?\mathrm{\Omega }}=\mathrm{0}\right.,\end{array}\right. where ,N\ge \mathrm{3},f:[r_{\mathrm{1}},r_{\mathrm{2}}]\times \mathbf{R}\times {\mathbf{R}}^{+}\to \mathbf{R} is continuous. In the general case where the nonlinear term f is not assumed to be non-negative, when f(r,u,\eta ) satisfies Nagumo-type condition on \eta, the existence of radial solutions is proved for the problem by using the method of lower and upper solutions and a truncating functional technique. Further, the uniqueness of radial solution is demonstrated under certain conditions.