Continuous spectrum for a class of nonhomogeneous differential operators

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.