Variational properties of the first curve of the Fu\v{c}\'{\i}k spectrum for elliptic operators

TL;DR

This paper introduces a new variational characterization of the first nontrivial Fučík spectrum curve for elliptic operators, analyzes its properties and asymptotic behavior, and compares it with other spectrum curves.

Contribution

It provides a novel variational approach to characterize the first Fučík spectrum curve and explores its asymptotic properties and relation to other spectrum curves.

Findings

The first Fučík spectrum curve is asymptotic to specific lines.

The new characterization distinguishes the first curve from other spectrum curves.

All spectrum curves share the same asymptotic lines but are distinct from each other.

Abstract

In this paper we present a new variational characteriztion of the first nontrival curve of the Fu\v{c}\'{\i}k spectrum for elliptic operators with Dirichlet boundary conditions. Moreover, we describe the asymptotic behaviour and some properties of this curve and of the corresponding eigenfunctions. In particular, this new characterization allows us to compare the first curve of the Fu\v{c}\'{\i}k spectrum with the infinitely many curves we obtained in previous works (see R. Molle, D. Passaseo, New properties of the Fu\v{c}\'{\i}k spectrum. C. R. Math. Acad. Sci. Paris 351 (2013), no. 17/18, 681--685 and R. Molle, D. Passaseo, Infinitely many new curves of the Fu\v{c}\'{\i}k spectrum. Ann. I. H. Poincar\'e - AN (2014), http://dx.doi.org/10.1016/j.anihpc.2014.05.007): for example, we show that these curves are all asymptotic to the same lines as the first curve, but they are all distinct from such a curve.