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

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.