manuscripta mathematica

, Volume 125, Issue 2, pp 157–167

Continuous spectrum for a class of nonhomogeneous differential operators

Article

DOI: 10.1007/s00229-007-0137-8

Cite this article as:
Mihăilescu, M. & Rădulescu, V. manuscripta math. (2008) 125: 157. doi:10.1007/s00229-007-0137-8

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 p1, p2, and q satisfy 1 < p2(x) < q(x) < p1(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.

Mathematics Subject Classification (2000)

35D0535J6035J7058E0568T4076A02

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of CraiovaCraiovaRomania
  2. 2.Institute of Mathematics “Simion Stoilow” of the Romanian AcademyBucharestRomania