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

MIHĂILESCU, MIHAI ; MOROŞANU, GHEORGHE

Glasgow mathematical journal, Tome 52 (2010) no. 3, pp. 517-527

Résumé

We study the eigenvalue problem = 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.

Détail

DOI : 10.1017/S001708951000039X

Mots-clés : 35D05, 35J60, 35J70, 58E05

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