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Nontrivial solutions of singular superlinear Sturm–Liouville problems. (English) Zbl 1100.34019

The authors study the singular superlinear problem

-(p(x)y ' ) ' -q(x)y=h(x)f(y),0<x<1,
α 1 y(0)+β 1 y ' (0)=0,α 2 y(1)+β 2 y ' (1)=0·

The function h is allowed to be singular at both x=0 and x=1. In addition, f is not assumed to be nonnegative. The assumption of nonnegativity of f has been very often required in the existing literature. Omitting this condition requires a different approach. Using topological degree theory, the authors establish conditions guaranteeing the existence of nontrivial solutions and positive solutions to the above boundary value problem. A nonsingular case is discussed as well.

34B18Positive solutions of nonlinear boundary value problems for ODE
34B16Singular nonlinear boundary value problems for ODE