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In this paper we present a new variational characteriztion of the firstnontrival curve of the Fu\v{c}\'{\i}k spectrum for elliptic operators withDirichlet boundary conditions. Moreover, we describe the asymptotic behaviourand some properties of this curve and of the corresponding eigenfunctions. Inparticular, this new characterization allows us to compare the first curve ofthe Fu\v{c}\'{\i}k spectrum with the infinitely many curves we obtained inprevious works (see R. Molle, D. Passaseo, New properties of the Fu\v{c}\'{\i}kspectrum. 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 thatthese curves are all asymptotic to the same lines as the first curve, but theyare all distinct from such a curve.

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