Partial differential equations
New properties of the Fučík spectrum
Comptes Rendus. Mathématique, Volume 351 (2013) no. 17-18, pp. 681-685.

In this Note we present some results on the Fučík spectrum for the Laplace operator, that give new information on its structure. In particular, these results show that, if Ω is a bounded domain of RN with N>1, then the Fučík spectrum has infinitely many curves asymptotic to the lines {λ1}×R and R×{λ1}, where λ1 denotes the first eigenvalue of the operator −Δ in H01(Ω). Notice that the situation is quite different in the case N=1; in fact, in this case, the Fučík spectrum may be obtained by direct computation and one can verify that it includes only two curves asymptotic to these lines. The method we use for the proof is completely variational.

Dans cette Note, nous présentons des résultats qui donnent de nouvelles informations sur la structure du spectre de Fučík pour lʼoperateur de Laplace. En particulier, ces résultats montrent que, si Ω est un domaine borné de RN avec N>1, alors le spectre de Fučík a un nombre infini de courbes qui ont comme asymptotes les droites {λ1}×R et R×{λ1}, où λ1 est la première valeur propre de lʼoperateur −Δ in H01(Ω). La situation est bien différente dans le cas N=1 ; en effect, dans ce cas, on peut vérifier quʼil y a seulement deux courbes dans le spectre de Fučík, qui ont ces droites comme asymptotes. La méthode de démonstration que nous avons suivie est complètement variationnelle.

Published online:
DOI: 10.1016/j.crma.2013.09.005
Molle, Riccardo 1; Passaseo, Donato 2

1 Dipartimento di Matematica, Università di Roma “Tor Vergata”, via della Ricerca Scientifica, 1, 00133 Roma, Italy
2 Dipartimento di Matematica “E. De Giorgi”, Università di Lecce, P.O. Box 193, 73100 Lecce, Italy
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Molle, Riccardo; Passaseo, Donato. New properties of the Fučík spectrum. Comptes Rendus. Mathématique, Volume 351 (2013) no. 17-18, pp. 681-685. doi : 10.1016/j.crma.2013.09.005.

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Work supported by the Italian national research project “Metodi variazionali e topologici nello studio di fenomeni non lineari”.