Infinitely many new curves of the Fučík spectrum

Molle, Riccardo; Passaseo, Donato

Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) no. 6, pp. 1145-1171.

Nous présentons des résultats qui donnent de nouvelles informations sur la structure du spectre de Fučík pour l'opérateur de Laplace. En particulier, ces résultats montrent que, si Ω est un domaine borné de ${ℝ}^N$ avec $N>1$, alors le spectre de Fučík a un nombre infini de courbes qui ont comme asymptotes les droites $\left\{{\lambda }_1\right\}\times ℝ$ et $ℝ\times \left\{{\lambda }_1\right\}$, où ${\lambda }_1$ est la première valeur propre de l'operateur −Δ in $H_0^1\left(\Omega \right)$. La situation est bien différente dans le cas $N=1$ ; en effect, dans ce cas on peut vérifier qu'il y a seulement deux courbes dans le spectre de Fučík, qui ont ces droites comme asymptotes.

In this paper 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 ${ℝ}^N$ with $N>1$, then the Fučík spectrum has infinitely many curves asymptotic to the lines $\left\{{\lambda }_1\right\}\times ℝ$ and $ℝ\times \left\{{\lambda }_1\right\}$, where ${\lambda }_1$ denotes the first eigenvalue of the operator −Δ in $H_0^1\left(\Omega \right)$. 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.

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