Some existence results for variational inequalities with nonlocal fractional operators
Autor(a) principal: | |
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Data de Publicação: | 2019 |
Outros Autores: | |
Tipo de documento: | Artigo |
Idioma: | eng |
Título da fonte: | Repositório Institucional da UNESP |
Texto Completo: | http://dx.doi.org/10.1016/j.na.2019.06.020 http://hdl.handle.net/11449/189381 |
Resumo: | In this paper we consider the following nonlocal fractional variational inequality u∈X0 s(Ω),u⩽ψa.e. in Ω,〈u,v−u〉X0 s(Ω)−λ〈u,v−u〉2⩾∫Ωfx,u(x),(−Δ)βu(x)(v(x)−u(x))dxaaaaaaaaaaaaaaaaaaaaaaaaaaaaafor anyv∈X0 s(Ω),v⩽ψa.e. in Ω, where Ω⊂RN is a smooth bounded open set with continuous boundary ∂Ω, s∈(0,1), N>2s, λ is a real parameter, f is function with subcritical growth, β∈(0,s∕2) and ψ is the obstacle function. As it is well-known, the dependence of the nonlinearity f on the term (−Δ)βu makes non-variational the nature of this problem. Using an iterative technique and a penalization method, we get the existence of a nontrivial nonnegative solution for the problem under consideration, performing the Mountain Pass Theorem. This result can be seen as the extension of known existence theorem for variational inequalities driven by the Laplace operator (or more general uniformly elliptic operators) to the nonlocal fractional setting. |
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Some existence results for variational inequalities with nonlocal fractional operatorsFractional LaplacianPenalization methodVariational inequalitiesVariational methodsIn this paper we consider the following nonlocal fractional variational inequality u∈X0 s(Ω),u⩽ψa.e. in Ω,〈u,v−u〉X0 s(Ω)−λ〈u,v−u〉2⩾∫Ωfx,u(x),(−Δ)βu(x)(v(x)−u(x))dxaaaaaaaaaaaaaaaaaaaaaaaaaaaaafor anyv∈X0 s(Ω),v⩽ψa.e. in Ω, where Ω⊂RN is a smooth bounded open set with continuous boundary ∂Ω, s∈(0,1), N>2s, λ is a real parameter, f is function with subcritical growth, β∈(0,s∕2) and ψ is the obstacle function. As it is well-known, the dependence of the nonlinearity f on the term (−Δ)βu makes non-variational the nature of this problem. Using an iterative technique and a penalization method, we get the existence of a nontrivial nonnegative solution for the problem under consideration, performing the Mountain Pass Theorem. This result can be seen as the extension of known existence theorem for variational inequalities driven by the Laplace operator (or more general uniformly elliptic operators) to the nonlocal fractional setting.Departamento de Matemática e Computação Fac. de Ciências e Tecnologia Universidade Estadual Paulista - UNESPDipartimento di Scienze Pure e Applicate (DiSPeA) Università degli Studi di Urbino Carlo Bo Piazza della Repubblica 13Departamento de Matemática e Computação Fac. de Ciências e Tecnologia Universidade Estadual Paulista - UNESPUniversidade Estadual Paulista (Unesp)Piazza della Repubblica 13Pimenta, Marcos Tadeu Oliveira [UNESP]Servadei, Raffaella2019-10-06T16:38:46Z2019-10-06T16:38:46Z2019-12-01info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articlehttp://dx.doi.org/10.1016/j.na.2019.06.020Nonlinear Analysis, Theory, Methods and Applications, v. 189.0362-546Xhttp://hdl.handle.net/11449/18938110.1016/j.na.2019.06.0202-s2.0-850685983550319425297974158Scopusreponame:Repositório Institucional da UNESPinstname:Universidade Estadual Paulista (UNESP)instacron:UNESPengNonlinear Analysis, Theory, Methods and Applicationsinfo:eu-repo/semantics/openAccess2024-06-19T14:31:49Zoai:repositorio.unesp.br:11449/189381Repositório InstitucionalPUBhttp://repositorio.unesp.br/oai/requestopendoar:29462024-08-05T14:21:10.079469Repositório Institucional da UNESP - Universidade Estadual Paulista (UNESP)false |
dc.title.none.fl_str_mv |
Some existence results for variational inequalities with nonlocal fractional operators |
title |
Some existence results for variational inequalities with nonlocal fractional operators |
spellingShingle |
Some existence results for variational inequalities with nonlocal fractional operators Pimenta, Marcos Tadeu Oliveira [UNESP] Fractional Laplacian Penalization method Variational inequalities Variational methods |
title_short |
Some existence results for variational inequalities with nonlocal fractional operators |
title_full |
Some existence results for variational inequalities with nonlocal fractional operators |
title_fullStr |
Some existence results for variational inequalities with nonlocal fractional operators |
title_full_unstemmed |
Some existence results for variational inequalities with nonlocal fractional operators |
title_sort |
Some existence results for variational inequalities with nonlocal fractional operators |
author |
Pimenta, Marcos Tadeu Oliveira [UNESP] |
author_facet |
Pimenta, Marcos Tadeu Oliveira [UNESP] Servadei, Raffaella |
author_role |
author |
author2 |
Servadei, Raffaella |
author2_role |
author |
dc.contributor.none.fl_str_mv |
Universidade Estadual Paulista (Unesp) Piazza della Repubblica 13 |
dc.contributor.author.fl_str_mv |
Pimenta, Marcos Tadeu Oliveira [UNESP] Servadei, Raffaella |
dc.subject.por.fl_str_mv |
Fractional Laplacian Penalization method Variational inequalities Variational methods |
topic |
Fractional Laplacian Penalization method Variational inequalities Variational methods |
description |
In this paper we consider the following nonlocal fractional variational inequality u∈X0 s(Ω),u⩽ψa.e. in Ω,〈u,v−u〉X0 s(Ω)−λ〈u,v−u〉2⩾∫Ωfx,u(x),(−Δ)βu(x)(v(x)−u(x))dxaaaaaaaaaaaaaaaaaaaaaaaaaaaaafor anyv∈X0 s(Ω),v⩽ψa.e. in Ω, where Ω⊂RN is a smooth bounded open set with continuous boundary ∂Ω, s∈(0,1), N>2s, λ is a real parameter, f is function with subcritical growth, β∈(0,s∕2) and ψ is the obstacle function. As it is well-known, the dependence of the nonlinearity f on the term (−Δ)βu makes non-variational the nature of this problem. Using an iterative technique and a penalization method, we get the existence of a nontrivial nonnegative solution for the problem under consideration, performing the Mountain Pass Theorem. This result can be seen as the extension of known existence theorem for variational inequalities driven by the Laplace operator (or more general uniformly elliptic operators) to the nonlocal fractional setting. |
publishDate |
2019 |
dc.date.none.fl_str_mv |
2019-10-06T16:38:46Z 2019-10-06T16:38:46Z 2019-12-01 |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/article |
format |
article |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
http://dx.doi.org/10.1016/j.na.2019.06.020 Nonlinear Analysis, Theory, Methods and Applications, v. 189. 0362-546X http://hdl.handle.net/11449/189381 10.1016/j.na.2019.06.020 2-s2.0-85068598355 0319425297974158 |
url |
http://dx.doi.org/10.1016/j.na.2019.06.020 http://hdl.handle.net/11449/189381 |
identifier_str_mv |
Nonlinear Analysis, Theory, Methods and Applications, v. 189. 0362-546X 10.1016/j.na.2019.06.020 2-s2.0-85068598355 0319425297974158 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
Nonlinear Analysis, Theory, Methods and Applications |
dc.rights.driver.fl_str_mv |
info:eu-repo/semantics/openAccess |
eu_rights_str_mv |
openAccess |
dc.source.none.fl_str_mv |
Scopus reponame:Repositório Institucional da UNESP instname:Universidade Estadual Paulista (UNESP) instacron:UNESP |
instname_str |
Universidade Estadual Paulista (UNESP) |
instacron_str |
UNESP |
institution |
UNESP |
reponame_str |
Repositório Institucional da UNESP |
collection |
Repositório Institucional da UNESP |
repository.name.fl_str_mv |
Repositório Institucional da UNESP - Universidade Estadual Paulista (UNESP) |
repository.mail.fl_str_mv |
|
_version_ |
1808128349954899968 |