Positive Radial Solutions for Elliptic Equations with Nonlinear Gradient Terms on the Unit Ball

This paper deals with the existence of positive radial solutions of the elliptic equation with nonlinear gradient term -Δu=f(|x|,u,|∇u|),x∈Ω,u|∂Ω=0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u = f(|x|,\,u,\,|\nabla u|)\,,\qquad x\in \Omega \,,\qquad \qquad \\ u|_{\partial \Omega }=0\,, \end{array}\right. \end{aligned}$$\end{document}where Ω={x∈RN:|x|<1}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega =\{x\in \mathbb {R}^N:\;|x|<1\}$$\end{document}, N≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\ge 2$$\end{document}, f:[0,1]×R+×R+→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f:[0,\,1]\times \mathbb {R}^+\times \mathbb {R}^+ \rightarrow \mathbb {R}$$\end{document} are continuous, R+=[0,∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^+=[0,\,\infty )$$\end{document}. Under some inequality conditions, the existence results of positive radial solution are obtained. The proofs of the main results are based on the method of lower and upper solutions and truncating function technique.