Advances in Differential Equations

Existence of positive solutions in the superlinear case via coincidence degree: the Neumann and the periodic boundary value problems

Guglielmo Feltrin and Fabio Zanolin

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Abstract

We prove the existence of positive periodic solutions for the second order nonlinear equation $u'' + a(x) g(u) = 0$, where $g(u)$ has superlinear growth at zero and at infinity. The weight function $a(x)$ is allowed to change its sign. Necessary and sufficient conditions for the existence of nontrivial solutions are obtained. The proof is based on Mawhin's coincidence degree and applies also to Neumann boundary conditions. Applications are given to the search of positive solutions for a nonlinear PDE in annular domains and for a periodic problem associated to a non-Hamiltonian equation.

Article information

Source
Adv. Differential Equations Volume 20, Number 9/10 (2015), 937-982.

Dates
First available in Project Euclid: 23 June 2015

Permanent link to this document
https://projecteuclid.org/euclid.ade/1435064518

Mathematical Reviews number (MathSciNet)
MR3360396

Zentralblatt MATH identifier
1345.34031

Subjects
Primary: 34B18: Positive solutions of nonlinear boundary value problems 34B15: Nonlinear boundary value problems 34C25: Periodic solutions 47H11: Degree theory [See also 55M25, 58C30]

Citation

Feltrin, Guglielmo; Zanolin, Fabio. Existence of positive solutions in the superlinear case via coincidence degree: the Neumann and the periodic boundary value problems. Adv. Differential Equations 20 (2015), no. 9/10, 937--982. https://projecteuclid.org/euclid.ade/1435064518.


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