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
The aim of this paper is to review the main recent results about the dynamics of nonlinear partial differential equations describing flux-saturated transport mechanisms, eventually in combination with porous media flow and/or reactions terms. The result is a system characterized by the presence of wave fronts which move defining an interface. This can be used to model different process in applications in a variety of areas as developmental biology or astrophysics. The concept of solution and its properties (well-posedness in a bounded variation scenario, Rankine-Hugoniot and geometric conditions for jumps, regularity results, finite speed of propagation, . . . ), qualitative study of these fronts (traveling waves in particular) and application in morphogenesis cover the panorama of this review.
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