This paper investigates the topological structure of the set of the positive solutions of the one-dimensional quasilinear indefinite Neumann problem $$ -left({u'}/{sqrt{1+{u'}^2}} ight)' = lambda a(x) f(u) ; ; ext{in } (0,1), quad u'(0)=0,;u'(1)=0, $$ where $lambdain RR$ is a parameter, $ain L^infty(0,1)$ changes sign, and $f in C^1(RR)$ is positive in $(0,+infty)$. 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.

Global components of positive bounded variation solutions of a one-dimensional indefinite quasilinear Neumann problem

LOPEZ GOMEZ, JULIAN
;
Pierpaolo Omari
2019-01-01

Abstract

This paper investigates the topological structure of the set of the positive solutions of the one-dimensional quasilinear indefinite Neumann problem $$ -left({u'}/{sqrt{1+{u'}^2}} ight)' = lambda a(x) f(u) ; ; ext{in } (0,1), quad u'(0)=0,;u'(1)=0, $$ where $lambdain RR$ is a parameter, $ain L^infty(0,1)$ changes sign, and $f in C^1(RR)$ is positive in $(0,+infty)$. 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.
File in questo prodotto:
File Dimensione Formato  
JLG-PO_ANS.pdf

Accesso chiuso

Tipologia: Documento in Versione Editoriale
Licenza: Copyright Editore
Dimensione 3.51 MB
Formato Adobe PDF
3.51 MB Adobe PDF   Visualizza/Apri   Richiedi una copia
LGO BV bif 19.pdf

Open Access dal 17/05/2020

Tipologia: Bozza finale post-referaggio (post-print)
Licenza: Copyright Editore
Dimensione 634.4 kB
Formato Adobe PDF
634.4 kB Adobe PDF Visualizza/Apri
Pubblicazioni consigliate

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11368/2943706
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 12
  • ???jsp.display-item.citation.isi??? 11
social impact