## 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.
Reviewer: Pavel Rehak (Brno)

### MSC:

 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 34B16 Singular nonlinear boundary value problems for ordinary differential equations
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### References:

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