, Volume 125, Issue 2, pp 157-167
Date: 06 Oct 2007

Continuous spectrum for a class of nonhomogeneous differential operators

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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.