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

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.

    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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