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Nontrivial solutions of singular superlinear Sturm--Liouville problems. (English) Zbl 1100.34019
The authors study the singular superlinear problem $$-(p(x)y')'-q(x)y=h(x)f(y),\quad 0<x<1,$$ $$\alpha_1 y(0)+\beta_1 y'(0)=0,\quad \alpha_2 y(1)+\beta_2 y'(1)=0.$$ 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
Full Text: DOI
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