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 Fučik 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. 2002 Éditions scientifiques et médicales Elsevier SAS RÉSUMÉ.-Nous démontrons l'existence d'une première valeur propre non triviale pour un problème asymétrique avec poids faisant intervenir le laplacien (cf. (1.2) ci-dessous) ou plus généralement le p-laplacien (cf. (1.3) ci-dessous). Une première application consiste en la description du début du spectre de Fučik avec poids pour ces opérateurs. Une autre application concerne l'étude de la nonrésonance pour les problèmes (1.1) et (1.5) ci-dessous. Une caractéristique de nos conditions de nonrésonance est qu'elles font intervenir des valeurs propres avec poids, plutôt que des restrictions ponctuelles. 2002 Éditions scientifiques et médicales Elsevier SAS
Let ∆p denote the p-Laplacian operator and Ω be a bounded domain in R N . We consider the eigenvalue problem(Ω) for a potential V and a weight function m that may change sign and be unbounded. Therefore the functional to be minimized is indefinite and may be unbounded from below. The main feature here is the introduction of a value α(V, m) that guarantees the boundedness of the energy over the weighted sphere M = {u ∈ W 1,p 0 (Ω); Ω m|u| p dx = 1}. We show that the above equation has a principal eigenvalue if and only if either m ≥ 0 and α(V, m) > 0 or m changes sign and α(V, m) ≥ 0. The existence of further eigenvalues is also treated here, mainly a second eigenvalue (to the right) and their dependence with respect to V and m.
Mathematics Subject Classification (2000). 35J20, 35J70, 35P05, 35P30.
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