Existence of positive stationary solutions for a reaction-diffusion system

In this paper, we will establish some existence results of positive stationary solutions for a reaction-diffusion system {−u″+λu=p(t)φu+h(t)f(u,φ),0<t<1,−φ″=q(t)uα,0<t<1,u(0)=u(1)=0,φ(0)=φ(1)=0.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \textstyle\begin{cases} -u''+\lambda u=p(t)\varphi u+h(t)f(u,\varphi), \quad0< t< 1,\\ -\varphi''=q(t) u^{\alpha}, \quad0< t< 1,\\ u(0)=u(1)=0,\\ \varphi(0)=\varphi(1)=0. \end{cases} $$\end{document} The main method used here is the well-known fixed point theorem of cone expansion and compression.