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ON AN EIGENVALUE PROBLEM FOR AN ANISOTROPIC ELLIPTIC EQUATION INVOLVING VARIABLE EXPONENTS

Published online by Cambridge University Press:  25 August 2010

MIHAI MIHĂILESCU  and
GHEORGHE MOROŞANU
MIHAI MIHĂILESCU
Affiliation:
Department of Mathematics, Central European University, 1051 Budapest, Hungary Department of Mathematics, University of Craiova, 200585 Craiova, Romania e-mail: mmihailes@yahoo.com
GHEORGHE MOROŞANU
Affiliation:
Department of Mathematics, Central European University, 1051 Budapest, Hungary e-mail: Morosanug@ceu.hu
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Abstract

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We study the eigenvalue problem $\(-\sum_{i=1}^N\di\partial_{x_i}(|\di\partial_{x_i}u |^{p_i(x)-2}\di\partial_{x_i}u)$ = λ|u|q(x)−2u in Ω, u = 0 on ∂Ω, where Ω is a bounded domain in ℝN with smooth boundary ∂Ω, λ is a positive real number, and p1,⋅ ⋅ ⋅, pN, q are continuous functions satisfying the following conditions: 2 ≤ pi(x) < N, 1 < q(x) for all x ∈ Ω, i ∈ {1,. . .,N}; there exist j, k ∈ {1,. . .,N}, j ≠ k, such that pj ≡ q in Ω, q is independent of xj and maxΩq < minΩpk. The main result of this paper establishes the existence of two positive constants λ0 and λ1 with λ0 ≤ λ1 such that every λ ∈(λ1, ∞) is an eigenvalue, while no λ ∈ (0, λ0) can be an eigenvalue of the above problem.

Keywords

35D0535J6035J7058E05

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 52 , Issue 3 , September 2010 , pp. 517 - 527
Copyright
Copyright © Glasgow Mathematical Journal Trust 2010

References

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