Positive solutions of semilinear differential equations with singularities (Q1270021)

The authors deal with the existence of positive solutions to a second-order differential equation of the form (1): \(z''+g(t)f(z)=0\) with suitable boundary conditions and \(g\in L^{1}(0,1)\). The function \(f\) is supposed to satisfy either \(0\leq \lim\sup_{x\to 0} f(x)/x<a\) and \(b<\lim\inf_{x\to\infty} f(x)/x \leq \infty\) or \(0\leq \lim\sup_{x\to \infty} f(x)/x<a\) and \(b<\lim\inf_{x\to 0} f(x)/x \leq \infty\) for suitable \(a\) and \(b\). The main idea is to change (1) into a Hammerstein integral equation of the form \(z(t)=\int_{0}^{1} k(t,s) g(s) f(z(s))ds\). The compactness of \(A\) is proved. This allows one to apply fixed point index theory for compact maps. Two applications are given: the existence of eigenvalues to the equation \(z''+\lambda g(t)f(z)=0\) and the existence of positive radial solutions to the equation \(\Delta u+h(| x|)f(u)=0\).