On the Fucik spectrum with indefinite weights
by
Jean-Pierre Gossez
Universite Libre de Bruxelles
Coauthors: M. Arias (Granada), J. Campos (Granada), M. Cuesta (Calais)

In this talk we will present some recent results on the Fucik spectrum with indefinite weights for the p-laplacian. Our starting point will be the following asymmetric eigenvalue problem:

-\Deltapu=\lambda[m(x)(u+)p-1-n(x)(u-)p-1].

The existence of a first nontrivial eigenvalue will be derived through a mountain pass argument. This eigenvalue can be characterized as the first eigenvalue \lambda having a changing sign eigenfunction. A first application concerns the description of the beginning of the Fucik spectrum, namely the existence of several first curves (generally one in each quadrant). The asymptotic behaviour of these first curves depends on p, on the dimension N, and on the supports of the weights. Another application concerns the study of nonresonance for the problem

-\Deltapu=f(x, u).

One feature of our nonresonance conditions is that they involve conditions on eigenvalues with weights rather then pointwise restrictions on limits of f.

Date received: February 25, 2002