New properties of the Fučík spectrum

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 R^N with N>1, then the Fučík spectrum has infinitely many curves asymptotic to the lines {λ_1}×R and R×{λ_1}, where λ_1 denotes the first eigenvalue of the operator -Delta in H_0^1(Ω). 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. The method we use for the proof is completely variational