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Positive solutions for super-sublinear indefinite problems: High multiplicity results via coincidence degree


Authors: Alberto Boscaggin, Guglielmo Feltrin and Fabio Zanolin
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
Published electronically: August 3, 2017
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Abstract: We study the periodic boundary value problem associated with the second order non-linear equation

$\displaystyle u'' + \bigr {(} \lambda a^{+}(t) - \mu a^{-}(t) \bigr {)} g(u) = 0,$    

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.

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Additional Information

Alberto Boscaggin
Affiliation: Department of Mathematics, University of Torino, via Carlo Alberto 10, 10123 Torino, Italy
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
Email: fabio.zanolin@uniud.it

DOI: https://doi.org/10.1090/tran/6992
Keywords: Boundary value problems, positive solutions, indefinite weight, super-sublinear non-linearity, multiplicity results, symbolic dynamics, coincidence degree
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”
Article copyright: © Copyright 2017 American Mathematical Society

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