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Continuous spectrum for a class of nonhomogeneous differential operators

Authors:	Mihai Mihăilescu Vicenţiu Rădulescu

Institution:	(1) Department of Mathematics, University of Craiova, 200585 Craiova, Romania;(2) Institute of Mathematics “Simion Stoilow” of the Romanian Academy, P.O. Box 1-764, 014700 Bucharest, Romania

Abstract:	We study the boundary value problem $-{\rm div}((\|\nabla u\|^{p_1(x)-2}+\|\nabla u\|^{p_2(x)-2})\nabla u)=\lambda\|u\|^{q(x)-2}u$ in Ω, u = 0 on ∂Ω, where Ω is a bounded domain in $\mathbb{R}^N$ 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 $\max_{y\in\overline\Omega}q(y) < \frac{N p_2(x)}{N-p_2(x)}$ for any $x\in\overline\Omega$ . The main result of this paper establishes the existence of two positive constants λ₀ and λ₁ with λ₀ ≤ λ₁ such that any $\lambda\in\lambda_1,\infty)$ is an eigenvalue, while any $\lambda\in(0,\lambda_0)$ is not an eigenvalue of the above problem.

Keywords:	35D05 35J60 35J70 58E05 68T40 76A02
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