Ambrosetti-Prodi type result to a Neumann problem via a topological approach

Elisa Sovrano

doi:10.3934/dcdss.2018019

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2018, Volume 11, Issue 2: 345-355. Doi: 10.3934/dcdss.2018019

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Ambrosetti-Prodi type result to a Neumann problem via a topological approach

Elisa Sovrano

Department of Mathematics, Computer Science and Physics, University of Udine, via delle Scienze 206,33100 Udine, Italy

Received: January 31, 2017

Revised: April 30, 2017

Published: April 2018

Work partially supported by the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). Progetto di Ricerca 2016: "Problemi differenziali non lineari: esistenza, molteplicità e proprietà qualitative delle soluzioni".

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We prove an Ambrosetti-Prodi type result for a Neumann problem associated to the equation $u''+f(x, u(x))=μ$ when the nonlinearity has the following form:$f(x, u):=a(x)g(u)-p(x)$. The assumptions considered generalize the classical one, $f(x, u)\to+∞$ as $|u|\to+∞$, without requiring any uniformity condition in $x$. The multiplicity result which characterizes these kind of problems will be proved by means of the shooting method.

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Mathematics Subject Classification: 34B15.

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Figure 1. Numerical simulations for the Neumann problem $(\mathcal{P}_{\mu})$ defined as in Example

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