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Positive subharmonic solutions to superlinear ODEs with indefinite weight. (English) Zbl 1381.34059

The author reviews some recent results he has obtained in collaboration with A. Boscaggin and F. Zanolin on the existence and multiplicity of periodic solutions for equations of the type \[ u''+q(t)g(u)=0, \] where \(g(u)\) has a superlinear growth both at zero and at infinity, and \(q(t)\) is a sign-changing periodic function. As a typical example, \(g(u)=u^p\), with \(p>1\). The main interest is on positive periodic solutions, of both harmonic and subharmonic type. The proofs involve topological degree methods, and the use of some generalized version of the Poincaré-Birkhoff theorem.

MSC:

34C25 Periodic solutions to ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
47H11 Degree theory for nonlinear operators

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