[Nouveaux résultats sur le spectre de Fučík]
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 avec , alors le spectre de Fučík a un nombre infini de courbes qui ont comme asymptotes les droites et , où est la première valeur propre de lʼoperateur −Δ in . La situation est bien différente dans le cas ; 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.
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 with , then the Fučík spectrum has infinitely many curves asymptotic to the lines and , where denotes the first eigenvalue of the operator −Δ in . Notice that the situation is quite different in the case ; 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.
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@article{CRMATH_2013__351_17-18_681_0,
author = {Molle, Riccardo and Passaseo, Donato},
title = {New properties of the {Fu\v{c}{\'\i}k} spectrum},
journal = {Comptes Rendus. Math\'ematique},
pages = {681--685},
publisher = {Elsevier},
volume = {351},
number = {17-18},
year = {2013},
doi = {10.1016/j.crma.2013.09.005},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2013.09.005/}
}
TY - JOUR AU - Molle, Riccardo AU - Passaseo, Donato TI - New properties of the Fučík spectrum JO - Comptes Rendus. Mathématique PY - 2013 SP - 681 EP - 685 VL - 351 IS - 17-18 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2013.09.005/ DO - 10.1016/j.crma.2013.09.005 LA - en ID - CRMATH_2013__351_17-18_681_0 ER -
%0 Journal Article %A Molle, Riccardo %A Passaseo, Donato %T New properties of the Fučík spectrum %J Comptes Rendus. Mathématique %D 2013 %P 681-685 %V 351 %N 17-18 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2013.09.005/ %R 10.1016/j.crma.2013.09.005 %G en %F CRMATH_2013__351_17-18_681_0
Molle, Riccardo; Passaseo, Donato. New properties of the Fučík spectrum. Comptes Rendus. Mathématique, Tome 351 (2013) no. 17-18, pp. 681-685. doi : 10.1016/j.crma.2013.09.005. http://www.numdam.org/articles/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”.
