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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 gL 1 (0,1). The function f is supposed to satisfy either 0limsup x0 f(x)/x<a and b<liminf x f(x)/x or 0limsup x f(x)/x<a and b<liminf x0 f(x)/x for suitable a and b.

The main idea is to change (1) into a Hammerstein integral equation of the form z(t)= 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 '' +λg(t)f(z)=0 and the existence of positive radial solutions to the equation Δu+h(|x|)f(u)=0.

34B15Nonlinear boundary value problems for ODE
45G10Nonsingular nonlinear integral equations