Asymmetric elliptic problems with indefinite weights

Abstract

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

Weighted Steklov problem under nonresonance conditions

“…To fix one's ideas, problem (P f ) can be found in [4,7] where particular cases of weight were considered. Throughout this subsection, we work on gathering needed properties to apply a version of the classical "Mountain Pass Theorem" for a C 1 functional restricted to a C 1 manifold (see [1,8]). Our purpose is of course to obtain existence results for (P f ) and by doing so, extend some of the known results in [4,5,7].…”

The beginning of the Fucik spectrum for a Steklov problem

“…Lemma 2.1 (see [1] and [2]) Let E be a real Banach space and let M := {u ∈ E; g(u) = 1}, where g ∈ C 1 (E, R) and 1 is a regular value of g. Let f ∈ C 1 (E, R) and consider the restrictionf of f to M . Let u, v ∈ M with u = v and assume that…”

“…Let u be an eigenfunction in M sm,n associated to c(sm, n) and let γ be the path in M sm,n from ϕ sm to −ϕ n constructed from u as in the proof of Proposition 31 of [1]. The pathγ(t) := (…”

“…Ω ⊂ R N be a bounded domain with a Lipschitz continuous boundary, where N ≥ 2, m, n ∈ L q (∂Ω) with N −1 p−1 < q if 1 < p ≤ N and q ≥ 1 if p > N . We proved the existence of a first nonprincipal positive eigenvalue c(m, n) for (1).…”

“…We also recall some results concerning c(m, n) the first nonprincipal positive eigenvalue of (1). Several properties of the c(m, n) as a function of the weights m, n are investigated in Section 3: continuity, monotonicity and homogeneity.…”