ON AN EIGENVALUE PROBLEM FOR AN ANISOTROPIC ELLIPTIC EQUATION INVOLVING VARIABLE EXPONENTS
2010 ◽
Vol 52
(3)
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pp. 517-527
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AbstractWe 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.
2008 ◽
Vol 06
(01)
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pp. 83-98
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2019 ◽
Vol 9
(1)
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pp. 1130-1144
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2018 ◽
Vol 24
(2)
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pp. 569-578
2018 ◽
Vol 20
(07)
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pp. 1750074
2018 ◽
Vol 7
(1)
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pp. 77-83
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2014 ◽
Vol 16
(04)
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pp. 1350046
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