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Positive solutions of semilinear differential equations with singularities. (English) Zbl 0909.34013
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 $0\le \lim\sup_{x\to 0} f(x)/x<a$ and $b<\lim\inf_{x\to\infty} f(x)/x \le \infty$ or $0\le \lim\sup_{x\to \infty} f(x)/x<a$ and $b<\lim\inf_{x\to 0} f(x)/x \le \infty$ 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 $\Delta u+h(| x|)f(u)=0$.

34B15Nonlinear boundary value problems for ODE
45G10Nonsingular nonlinear integral equations
Full Text: DOI
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