Continuous spectrum for a class of nonhomogeneous differential operators

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• 作者：Mihai Mih?ilescu and Vicen?iu R?dulescu
• 关键词：Mathematics Subject Classification (2000) 35D05 ; 35J60 ; 35J70 ; 58E05 ; 68T40 ; 76A02
• 刊名：manuscripta mathematica
• 出版年：2008
• 出版时间：February, 2008
• 年：2008
• 卷：125
• 期：2
• 页码：157-167
• 全文大小：166.5 KB

文摘

We study the boundary value problem -div((|?u|p₁(x)-2+|?u|p₂(x)-2)?u)=l|u|q(x)-2u-{\rm div}((|\nabla u|^{p_1(x)-2}+|\nabla u|^{p_2(x)-2})\nabla u)=\lambda|u|^{q(x)-2}u in Ω, u = 0 on ∂Ω, where Ω is a bounded domain in \mathbbRN\mathbb{R}^N with smooth boundary, λ is a positive real number, and the continuous functions p ₁, p ₂, and q satisfy 1 < p ₂(x) < q(x) < p ₁(x) < N and max<sub>y ? [`(W)]</sub>q(y) < \fracN p<sub>2</sub>(x)N-p<sub>2</sub>(x)\max_{y\in\overline\Omega}q(y) for any x ? [`(W)]x\in\overline\Omega. The main result of this paper establishes the existence of two positive constants λ₀ and λ₁ with λ₀ ≤ λ₁ such that any l ? [l₁,£¤)\lambda\in[\lambda_1,\infty) is an eigenvalue, while any l ? (0,l₀)\lambda\in(0,\lambda_0) is not an eigenvalue of the above problem.