Some classes of elliptic problems with singular nonlinearities
Autor(a) principal: | |
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Data de Publicação: | 2016 |
Tipo de documento: | Tese |
Idioma: | por |
Título da fonte: | Repositório Institucional da UFPE |
Texto Completo: | https://repositorio.ufpe.br/handle/123456789/23433 |
Resumo: | Singular elliptic problems has been extensively studied and it has attracted the attention of many research in various contexts and applications. The purpose of this thesis is to study singular elliptic problems in riemannian manifolds. We investigate a semilinear elliptic problem involving singular nonlinearities and advection and we prove the existence of a parameter λ∗ > 0 such that for λ ∈ (0, λ∗) there exists a minimal classical solution which is semi-stable and for λ > λ∗ there are no solutions of any kind. Futhermore we obtain Lp estimates for minimal solutions uniformly in λ and determine the critical dimension for this class of problems. As an application, we prove that the extremal solution is classical whenever the dimension of the riemannian manifold is below the critical dimension. We analyse the branch of minimal solutions and we prove multiplicity of solutions close to extremal parameter. We also prove symmetry and monotonicity properties for the class of semi-stable solutions and we obtain L∞ estimates for the extremal solution. Moreover, we study a class of problems involving the p−Laplace Beltrami operator in a geodesic ball of a riemannian model and we establish L∞ and W 1,p estimates for semi-stable, radially symmetric and decreasing solutions. As an application we prove regularity results for extremal solution of a quasilinear elliptic problem with Dirichlet boundary conditions. In the last chapter we study an elliptic system and we prove the existence of a curve which splits the positive quadrant of the plane into two disjoint sets, where there is classical solution while in the other there is no solution. We establish upper and lower estimates for the critical curve and regularity results for solutions on this curve. |
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CLEMENTE, Rodrigo Genuinohttp://lattes.cnpq.br/4351609162717260http://lattes.cnpq.br/6069135199129029do Ó, João Marcos Bezerra2018-01-30T19:06:26Z2018-01-30T19:06:26Z2016-02-24https://repositorio.ufpe.br/handle/123456789/23433Singular elliptic problems has been extensively studied and it has attracted the attention of many research in various contexts and applications. The purpose of this thesis is to study singular elliptic problems in riemannian manifolds. We investigate a semilinear elliptic problem involving singular nonlinearities and advection and we prove the existence of a parameter λ∗ > 0 such that for λ ∈ (0, λ∗) there exists a minimal classical solution which is semi-stable and for λ > λ∗ there are no solutions of any kind. Futhermore we obtain Lp estimates for minimal solutions uniformly in λ and determine the critical dimension for this class of problems. As an application, we prove that the extremal solution is classical whenever the dimension of the riemannian manifold is below the critical dimension. We analyse the branch of minimal solutions and we prove multiplicity of solutions close to extremal parameter. We also prove symmetry and monotonicity properties for the class of semi-stable solutions and we obtain L∞ estimates for the extremal solution. Moreover, we study a class of problems involving the p−Laplace Beltrami operator in a geodesic ball of a riemannian model and we establish L∞ and W 1,p estimates for semi-stable, radially symmetric and decreasing solutions. As an application we prove regularity results for extremal solution of a quasilinear elliptic problem with Dirichlet boundary conditions. In the last chapter we study an elliptic system and we prove the existence of a curve which splits the positive quadrant of the plane into two disjoint sets, where there is classical solution while in the other there is no solution. We establish upper and lower estimates for the critical curve and regularity results for solutions on this curve.CAPESProblemas elípticos singulares têm sido extensivamente estudados nas últimas décadas. Nesta tese, abordamos classes de problemas não lineares modelados em variedades riemannianas. Investigamos inicialmente um problema elíptico semilinear envolvendo não linearidades singulares e advecção e provamos resultados de existência do parâmetro extremal λ∗ > 0 tal que para λ ∈ (0, λ∗) existe uma solução minimal clássica a qual é semiestável e para λ > λ∗ não existem soluções de nenhum tipo. Além disso, obtivemos estimativas Lp para as soluções minimais que são uniformes em λ e determinamos as dimensões críticas para esta classe de problemas. Como uma aplicação, provamos a regularidade da solução extremal quando a dimensão da variedade riemanniana está abaixo da dimensão crítica. Analisamos o ramo das soluções minimais e provamos multiplicidade de soluções próximo do λ∗. Provamos também simetria e monotonicidade para a classe das soluções semiestáveis e provamos estimativas L∞ para a solução extremal. Estudamos também uma classe de equações envolvendo o operador p−Laplace Beltrami em uma bola geodésica de uma variedade Riemanniana modelo e estabelecemos estimativas L∞ e W 1,p para soluções semiestáveis, radialmente simétricas e decrescentes. Como aplicação, provamos resultados de regularidade para soluções extremais para um problema quasilinear com condição de fronteira de Dirichlet. No último capítulo, estudamos um sistema elíptico e provamos a existência de uma curva que divide o primeiro quadrante do plano em dois conjuntos disjuntos, um dos quais existe solução clássica enquanto que no outro não existe solução. Estabelecemos estimativas superiores e inferiores para tal curva e resultados de regularidade para soluções sobre a curva.porUniversidade Federal de PernambucoPrograma de Pos Graduacao em MatematicaUFPEBrasilAttribution-NonCommercial-NoDerivs 3.0 Brazilhttp://creativecommons.org/licenses/by-nc-nd/3.0/br/info:eu-repo/semantics/openAccessMatemáticaAnálise não-linearEquações diferenciais parciaisSome classes of elliptic problems with singular nonlinearitiesinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/doctoralThesisdoutoradoreponame:Repositório Institucional da UFPEinstname:Universidade Federal de Pernambuco (UFPE)instacron:UFPETHUMBNAILtese Rodrigo.pdf.jpgtese Rodrigo.pdf.jpgGenerated Thumbnailimage/jpeg1283https://repositorio.ufpe.br/bitstream/123456789/23433/5/tese%20Rodrigo.pdf.jpg4bb5650cf9dbb5c501ffcad6d926215eMD55ORIGINALtese Rodrigo.pdftese Rodrigo.pdfapplication/pdf804512https://repositorio.ufpe.br/bitstream/123456789/23433/1/tese%20Rodrigo.pdfe2eb63d64e0fc9e621c117fa245490c3MD51CC-LICENSElicense_rdflicense_rdfapplication/rdf+xml; charset=utf-8811https://repositorio.ufpe.br/bitstream/123456789/23433/2/license_rdfe39d27027a6cc9cb039ad269a5db8e34MD52LICENSElicense.txtlicense.txttext/plain; charset=utf-82311https://repositorio.ufpe.br/bitstream/123456789/23433/3/license.txt4b8a02c7f2818eaf00dcf2260dd5eb08MD53TEXTtese Rodrigo.pdf.txttese Rodrigo.pdf.txtExtracted texttext/plain151896https://repositorio.ufpe.br/bitstream/123456789/23433/4/tese%20Rodrigo.pdf.txt7da5f7f5438fa0e7cc46c412e396400fMD54123456789/234332019-10-25 22:55:49.752oai:repositorio.ufpe.br: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Repositório InstitucionalPUBhttps://repositorio.ufpe.br/oai/requestattena@ufpe.bropendoar:22212019-10-26T01:55:49Repositório Institucional da UFPE - Universidade Federal de Pernambuco (UFPE)false |
dc.title.pt_BR.fl_str_mv |
Some classes of elliptic problems with singular nonlinearities |
title |
Some classes of elliptic problems with singular nonlinearities |
spellingShingle |
Some classes of elliptic problems with singular nonlinearities CLEMENTE, Rodrigo Genuino Matemática Análise não-linear Equações diferenciais parciais |
title_short |
Some classes of elliptic problems with singular nonlinearities |
title_full |
Some classes of elliptic problems with singular nonlinearities |
title_fullStr |
Some classes of elliptic problems with singular nonlinearities |
title_full_unstemmed |
Some classes of elliptic problems with singular nonlinearities |
title_sort |
Some classes of elliptic problems with singular nonlinearities |
author |
CLEMENTE, Rodrigo Genuino |
author_facet |
CLEMENTE, Rodrigo Genuino |
author_role |
author |
dc.contributor.authorLattes.pt_BR.fl_str_mv |
http://lattes.cnpq.br/4351609162717260 |
dc.contributor.advisorLattes.pt_BR.fl_str_mv |
http://lattes.cnpq.br/6069135199129029 |
dc.contributor.author.fl_str_mv |
CLEMENTE, Rodrigo Genuino |
dc.contributor.advisor1.fl_str_mv |
do Ó, João Marcos Bezerra |
contributor_str_mv |
do Ó, João Marcos Bezerra |
dc.subject.por.fl_str_mv |
Matemática Análise não-linear Equações diferenciais parciais |
topic |
Matemática Análise não-linear Equações diferenciais parciais |
description |
Singular elliptic problems has been extensively studied and it has attracted the attention of many research in various contexts and applications. The purpose of this thesis is to study singular elliptic problems in riemannian manifolds. We investigate a semilinear elliptic problem involving singular nonlinearities and advection and we prove the existence of a parameter λ∗ > 0 such that for λ ∈ (0, λ∗) there exists a minimal classical solution which is semi-stable and for λ > λ∗ there are no solutions of any kind. Futhermore we obtain Lp estimates for minimal solutions uniformly in λ and determine the critical dimension for this class of problems. As an application, we prove that the extremal solution is classical whenever the dimension of the riemannian manifold is below the critical dimension. We analyse the branch of minimal solutions and we prove multiplicity of solutions close to extremal parameter. We also prove symmetry and monotonicity properties for the class of semi-stable solutions and we obtain L∞ estimates for the extremal solution. Moreover, we study a class of problems involving the p−Laplace Beltrami operator in a geodesic ball of a riemannian model and we establish L∞ and W 1,p estimates for semi-stable, radially symmetric and decreasing solutions. As an application we prove regularity results for extremal solution of a quasilinear elliptic problem with Dirichlet boundary conditions. In the last chapter we study an elliptic system and we prove the existence of a curve which splits the positive quadrant of the plane into two disjoint sets, where there is classical solution while in the other there is no solution. We establish upper and lower estimates for the critical curve and regularity results for solutions on this curve. |
publishDate |
2016 |
dc.date.issued.fl_str_mv |
2016-02-24 |
dc.date.accessioned.fl_str_mv |
2018-01-30T19:06:26Z |
dc.date.available.fl_str_mv |
2018-01-30T19:06:26Z |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/doctoralThesis |
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doctoralThesis |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
https://repositorio.ufpe.br/handle/123456789/23433 |
url |
https://repositorio.ufpe.br/handle/123456789/23433 |
dc.language.iso.fl_str_mv |
por |
language |
por |
dc.rights.driver.fl_str_mv |
Attribution-NonCommercial-NoDerivs 3.0 Brazil http://creativecommons.org/licenses/by-nc-nd/3.0/br/ info:eu-repo/semantics/openAccess |
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Attribution-NonCommercial-NoDerivs 3.0 Brazil http://creativecommons.org/licenses/by-nc-nd/3.0/br/ |
eu_rights_str_mv |
openAccess |
dc.publisher.none.fl_str_mv |
Universidade Federal de Pernambuco |
dc.publisher.program.fl_str_mv |
Programa de Pos Graduacao em Matematica |
dc.publisher.initials.fl_str_mv |
UFPE |
dc.publisher.country.fl_str_mv |
Brasil |
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Universidade Federal de Pernambuco |
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