An asymmetric Neumann problem with weights

Abstract

We prove the existence of a first nonprincipal eigenvalue for an asymmetric Neumann problem with weights involving the p-Laplacian (cf. (1.2) below). As an application we obtain a first nontrivial curve in the corresponding Fučik spectrum (cf. (1.4) below). The case where one of the weights has meanvalue zero requires some special attention in connexion with the (PS) condition and with the mountain pass geometry.

Spectrum of One-Dimensional p-Laplacian with an Indefinite Integrable Weight

“…For positive weights ω ∈ W γ , principal eigenvalues of (1.1) are always 0. It has been observed in [2] that, besides the zero principal eigenvalue, for some indefinite weights, the Neumann problem may admit positive principal eigenvalues which are important in some applied problems [12]. In Theorem 4.3, we will use the quasi-monotonicity to give a complete characterization of positive principal eigenvalues of (1.1).…”

On the First Curve in the FučiK Spectrum with Weights for a Mixedp-Laplacian

“…Moreover, the asymptotic behavior of these first curves was shown to depend on the supports of the weights. The case of the Neumann boundary conditions was considered later in [2] where, contrary to what happened in the Dirichlet case, the asymptotic behavior of the first curve did not depend on the supports of the weights.…”

“…Our purpose in this work is to investigate a case which is somehow intermediate between Dirichlet's and Neumann's, that is, the case of the classical "mixed" boundary conditions. While trying to adapt the approach in [1,2] to the present situation, new difficulties arise in connection with the lack of regularity of the eigenfunctions. It is well known that weak solutions of degenerate elliptic quasilinear equations, more generally the one considered here, under Dirichlet or Neumann boundary conditions are essentially bounded in Ω and at least of class C α loc (Ω) (cf.…”

On the Fučík spectrum for equations with symmetries

“…The case N = 1 was treated by Drábek [14]; in the N -dimensional case, Cuesta-de Figueiredo-Gossez [10] characterized the first curve in the Fučík spectrum, Perera [28] considered the general case (higher order branches), Arias-Campos-CuestaGossez treated the case with a and b variable [3] and indefinite [4].…”