Positive solutions for super-sublinear indefinite problems: High multiplicity results via coincidence degree
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- by Alberto Boscaggin, Guglielmo Feltrin and Fabio Zanolin PDF
- Trans. Amer. Math. Soc. 370 (2018), 791-845 Request permission
Abstract:
We study the periodic boundary value problem associated with the second order non-linear equation \begin{equation*} u'' + \bigr {(} \lambda a^{+}(t) - \mu a^{-}(t) \bigr {)} g(u) = 0, \end{equation*} where $g(u)$ has superlinear growth at zero and sublinear growth at infinity. For $\lambda , \mu$ positive and large, we prove the existence of $3^{m}-1$ positive $T$-periodic solutions when the weight function $a(t)$ has $m$ positive humps separated by $m$ negative ones (in a $T$-periodicity interval). As a byproduct of our approach we also provide an abundance of positive subharmonic solutions and symbolic dynamics. The proof is based on coincidence degree theory for locally compact operators on open unbounded sets and also applies to Neumann and Dirichlet boundary conditions. Finally, we deal with radially symmetric positive solutions for the Neumann and the Dirichlet problems associated with elliptic PDEs.References
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Additional Information
- Alberto Boscaggin
- Affiliation: Department of Mathematics, University of Torino, via Carlo Alberto 10, 10123 Torino, Italy
- MR Author ID: 896012
- Email: alberto.boscaggin@unito.it
- Guglielmo Feltrin
- Affiliation: SISSA - International School for Advanced Studies, via Bonomea 265, 34136 Trieste, Italy
- Address at time of publication: Department of Mathematics, University of Mons, place du Parc 20, B-7000 Mons, Belgium
- Email: guglielmo.feltrin@sissa.it, guglielmo.feltrin@umons.ac.be
- Fabio Zanolin
- Affiliation: Department of Mathematics, Computer Science and Physics, University of Udine, via delle Scienze 206, 33100 Udine, Italy
- MR Author ID: 186545
- Email: fabio.zanolin@uniud.it
- Received by editor(s): December 18, 2015
- Received by editor(s) in revised form: April 21, 2016
- Published electronically: August 3, 2017
- Additional Notes: This work was performed under the auspicies of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The second and third authors were partially supported by the GNAMPA Project 2015 “Problemi al contorno associati ad alcune classi di equazioni differenziali non lineari”. The first author was partially supported by the GNAMPA Project 2015 “Equazioni Differenziali Ordinarie sulla retta reale” and by the project ERC Advanced Grant 2013 n. 339958 “Complex Patterns for Strongly Interacting Dynamical Systems - COMPAT”
- © Copyright 2017 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 370 (2018), 791-845
- MSC (2010): Primary 34B15, 34B18, 34C25, 34C28, 47H11
- DOI: https://doi.org/10.1090/tran/6992
- MathSciNet review: 3729488