The aim of this paper is to study the spectrum of the fourth order eigenvalue boundary value problem ⎧Δ²u = αu + βΔu in Ω, ⎨ ⎩u = Δu = 0 on ∂Ω. where (α,β) ∈ ℝ². We prove the existence of a first nontrivial curve of this spectrum and we give its variational characterization. Moreover we prove some properties of this curve, e.g., continuity, convexity, and asymptotic behavior. As an application, we study the non-resonance of solutions below the first principal eigencurve of the biharmonic problem ⎧Δ²u = f(u,x) + βΔu + h in Ω, ⎨ ⎩Δu = u = 0 ∂Ω, where f: Ω × ℝ → ℝ is a Carathéodory function and h is a given function in L²(Ω).
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-am41-2-14,
author = {Omar Chakrone and Najib Tsouli and Mostafa Rahmani and Omar Darhouche},
title = {Boundary eigencurve problems involving the biharmonic operator},
journal = {Applicationes Mathematicae},
volume = {41},
year = {2014},
pages = {267-275},
zbl = {1304.35464},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-am41-2-14}
}
Omar Chakrone; Najib Tsouli; Mostafa Rahmani; Omar Darhouche. Boundary eigencurve problems involving the biharmonic operator. Applicationes Mathematicae, Tome 41 (2014) pp. 267-275. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-am41-2-14/