The behaviour of the p (x)-Laplacian eigenvalue problem as p (x) → ∞
In this paper we study the behaviour of the solutions to the eigenvalue problem corresponding to the p (x)-Laplacian operator{(- div (| ∇ u |p (x) - 2 ∇ u) = Λp (x) | u |p (x) - 2 u,, in Ω,; u = 0,, on ∂ Ω,) as p (x) → ∞. We consider a sequence of functions pn (x) that goes to infinity uniformly in...
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paper:paper_0022247X_v363_n2_p502_PerezLlanos2023-06-08T14:47:53Z The behaviour of the p (x)-Laplacian eigenvalue problem as p (x) → ∞ Rossi, Julio Daniel Eigenvalue problems p (x)-Laplacian ∞-Laplacian In this paper we study the behaviour of the solutions to the eigenvalue problem corresponding to the p (x)-Laplacian operator{(- div (| ∇ u |p (x) - 2 ∇ u) = Λp (x) | u |p (x) - 2 u,, in Ω,; u = 0,, on ∂ Ω,) as p (x) → ∞. We consider a sequence of functions pn (x) that goes to infinity uniformly in over(Ω, -). Under adequate hypotheses on the sequence pn, namely that the limits∇ ln pn (x) → ξ (x), and frac(pn, n) (x) → q (x) exist, we prove that the corresponding eigenvalues Λpn and eigenfunctions upn verify that(Λpn)1 / n → Λ∞, upn → u∞ uniformly in over(Ω, -), where Λ∞, u∞ is a nontrivial viscosity solution of the following problem{(min {- Δ∞ u∞ - | ∇ u∞ |2 log (| ∇ u∞ |) 〈 ξ, ∇ u∞ 〉, | ∇ u∞ |q - Λ∞ u∞ q} = 0, in Ω,; u∞ = 0, on ∂ Ω .). © 2009 Elsevier Inc. All rights reserved. Fil:Rossi, J.D. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. 2010 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_0022247X_v363_n2_p502_PerezLlanos http://hdl.handle.net/20.500.12110/paper_0022247X_v363_n2_p502_PerezLlanos |
| institution |
Universidad de Buenos Aires |
| institution_str |
I-28 |
| repository_str |
R-134 |
| collection |
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) |
| topic |
Eigenvalue problems p (x)-Laplacian ∞-Laplacian |
| spellingShingle |
Eigenvalue problems p (x)-Laplacian ∞-Laplacian Rossi, Julio Daniel The behaviour of the p (x)-Laplacian eigenvalue problem as p (x) → ∞ |
| topic_facet |
Eigenvalue problems p (x)-Laplacian ∞-Laplacian |
| description |
In this paper we study the behaviour of the solutions to the eigenvalue problem corresponding to the p (x)-Laplacian operator{(- div (| ∇ u |p (x) - 2 ∇ u) = Λp (x) | u |p (x) - 2 u,, in Ω,; u = 0,, on ∂ Ω,) as p (x) → ∞. We consider a sequence of functions pn (x) that goes to infinity uniformly in over(Ω, -). Under adequate hypotheses on the sequence pn, namely that the limits∇ ln pn (x) → ξ (x), and frac(pn, n) (x) → q (x) exist, we prove that the corresponding eigenvalues Λpn and eigenfunctions upn verify that(Λpn)1 / n → Λ∞, upn → u∞ uniformly in over(Ω, -), where Λ∞, u∞ is a nontrivial viscosity solution of the following problem{(min {- Δ∞ u∞ - | ∇ u∞ |2 log (| ∇ u∞ |) 〈 ξ, ∇ u∞ 〉, | ∇ u∞ |q - Λ∞ u∞ q} = 0, in Ω,; u∞ = 0, on ∂ Ω .). © 2009 Elsevier Inc. All rights reserved. |
| author |
Rossi, Julio Daniel |
| author_facet |
Rossi, Julio Daniel |
| author_sort |
Rossi, Julio Daniel |
| title |
The behaviour of the p (x)-Laplacian eigenvalue problem as p (x) → ∞ |
| title_short |
The behaviour of the p (x)-Laplacian eigenvalue problem as p (x) → ∞ |
| title_full |
The behaviour of the p (x)-Laplacian eigenvalue problem as p (x) → ∞ |
| title_fullStr |
The behaviour of the p (x)-Laplacian eigenvalue problem as p (x) → ∞ |
| title_full_unstemmed |
The behaviour of the p (x)-Laplacian eigenvalue problem as p (x) → ∞ |
| title_sort |
behaviour of the p (x)-laplacian eigenvalue problem as p (x) → ∞ |
| publishDate |
2010 |
| url |
https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_0022247X_v363_n2_p502_PerezLlanos http://hdl.handle.net/20.500.12110/paper_0022247X_v363_n2_p502_PerezLlanos |
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AT rossijuliodaniel thebehaviourofthepxlaplacianeigenvalueproblemaspx AT rossijuliodaniel behaviourofthepxlaplacianeigenvalueproblemaspx |
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1768546616394907648 |