Asymmetric elliptic problems with indefinite weights

We prove the existence of a first nontrivial eigenvalue for an asymmetric elliptic problem with weights involving the laplacian (cf. (1.2) below) or more generally the p-laplacian (cf. (1.3) below). A first application is given to the description of the beginning of the Fucik spectrum with weights for these operators. Another application concerns the study of nonresonance for the problems (1.1) and (1.5) below. One feature of our nonresonance conditions is that they involve eigenvalues with weights instead of pointwise restrictions. (C) 2002 Editions scientifiques et medicales Elsevier SAS.