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)


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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