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Global Components of Positive Bounded Variation Solutions of a One-Dimensional Indefinite Quasilinear Neumann Problem

Julian López-Gómez and Pierpaolo Omari ORCID logo

Abstract

This paper investigates the topological structure of the set of the positive solutions of the one-dimensional quasilinear indefinite Neumann problem

{-(u1+u2)=λa(x)f(u)in (0,1),u(0)=0,u(1)=0,

where λ is a parameter, aL(0,1) changes sign, and fC1() is positive in (0,+). The attention is focused on the case f(0)=0 and f(0)=1, where we can prove, likely for the first time in the literature, a bifurcation result for this problem in the space of bounded variation functions. Namely, the existence of global connected components of the set of the positive solutions, emanating from the line of the trivial solutions at the two principal eigenvalues of the linearized problem around 0, is established. The solutions in these components are regular, as long as they are small, while they may develop jump singularities at the nodes of the weight function a, as they become larger, thus showing the possible coexistence along the same component of regular and singular solutions.

MSC 2010: 35J93; 34B18; 35B32

Communicated by Chris Cosner


Funding source:

Funding statement: The authors have been supported by “Università degli Studi di Trieste-Finanziamento di Ateneo per Progetti di Ricerca Scientifica-FRA 2015” and by the INdAM-GNAMPA 2017 Research Project “Problemi fortemente nonlineari: esistenza, molteplicità, regolarità delle soluzioni”. This paper has been written under the auspices of the Ministry of Science, Technology and Universities of Spain under Research Grants MTM2015-65899-P and PGC2018-097104-B-100.

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Received: 2019-03-11
Accepted: 2019-04-26
Published Online: 2019-05-16
Published in Print: 2019-08-01

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