Abstract
abstract: In this paper we establish the existence of the second eigencuve of the p-laplacian with indefinite weights, we obtain also their asymptotic behavior and variational formulation.
Highlights
Moussaoui abstract: In this paper we establish the existence of the second eigencurves of the p-laplacian with indefinite weights. we obtain their asymptotic behavior and variational formulation
The purpose of this article is to study the following problem: For α ∈ R Find the existence of real numbers β(α) such that (α, β(α)) ∈ C2 and the asymptotic behavior of β(α) as |α| → +∞, Many results have been obtained on this kind of problems (see; [4], [5], [8], in [5] the authors proved some properties related to the first eigencurve C1 such as concavity, differentiability and the asymptotic behavior, this last property can not be adapted to the other eigencurves, in [8] the authors have studied this class of problems under the following assumptions m, m′ ∈ M +(Ω) and ess infΩ m′ > 0
Proof: For each α < 0 there exists β2(α) > 0 such that λ2(αm + β2(α)m′ ) = 1 (see theorem 3.3), we have αm + β2(α)m′ > 0 in Ωα with meas (Ωα) > 0, so necessarily Ωα ⊂ Ω⋆m′ , we deduce that: lim inf α→−∞
Summary
Proof: To prove the first result, we consider the real function hα(.) defined by hα(t) = λ2(αm+tm′), hα is well defined and continuous in [0, +∞[ (see proposition 2.1), on the other hand, if α ∈]0, λ2(m)], we have: hα(0) = λ2(αm) λ2 (m) α and for t > 0, we have: hα(t) = λ2(αm + tm′ ) 3. If meas (Ω−m) = 0, for each α ∈ R⋆−, there exists a unique β2(α) ∈ R such that (α, β2(α)) ∈ C2.
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