Hénon type equations with nonlinearities in the critical growth range
Autor(a) principal: | |
---|---|
Data de Publicação: | 2017 |
Tipo de documento: | Tese |
Idioma: | eng |
Título da fonte: | Repositório Institucional da UFPE |
Texto Completo: | https://repositorio.ufpe.br/handle/123456789/25399 |
Resumo: | In this work, using variational methods we have investigated the existence of solutions for some Hénon type equations, which are characterized by the presence of the weight lxlᵅ in the nonlinearity with α > 0. When we are working in the radial context, this characteristic modifies the critical growth of the nonlinearities in some senses. This fact allows us to study some well-known problems under new perspectives. For this purpose, we have considered three different classes of problems with critical nonlinearity which presents the weight of Hénon. Firstly, we have studied the class of problem with a Trudinger- Moser nonlinearity in critical range in R². In the subcritical case, there was no diference if we have looked for weak solutions in H¹₀ (B₁) or in H¹₀,rad(B₁). Nevertheless, in the critical case we have needed to adapt some hypotheses when we have changed the space where we were seeking the solutions. For the second problem, we have kept working with exponential nonlinearity in R², but we were treating an Ambrosseti-Prodi problem for which we have searched two weak solutions. In the subcritical case, analogously to first problem, the radially symmetric solutions were obtained as the solutions in H¹₀ (B₁), what have not happened in the critical case. Thus, again some assumptions have had to depend on the context where we were searching for the solutions. Lastly, we have studied a natural version of the second problem with the nonlinearity involving critical Sobolev growth in Rᴺ (N ≥ 3). In this last problem, we have searched the existence of solutions only in the radial critical case because the others cases were almost identical to problems with nonlinearities without the weight of Hénon. |
id |
UFPE_d275118958e3ef4a50113d61fc2a4da8 |
---|---|
oai_identifier_str |
oai:repositorio.ufpe.br:123456789/25399 |
network_acronym_str |
UFPE |
network_name_str |
Repositório Institucional da UFPE |
repository_id_str |
2221 |
spelling |
BARBOZA, Eudes Mendeshttp://lattes.cnpq.br/9426464458648172http://lattes.cnpq.br/6069135199129029DO Ó, Joao Marcos Bezerra2018-08-06T17:23:24Z2018-08-06T17:23:24Z2017-05-30https://repositorio.ufpe.br/handle/123456789/25399In this work, using variational methods we have investigated the existence of solutions for some Hénon type equations, which are characterized by the presence of the weight lxlᵅ in the nonlinearity with α > 0. When we are working in the radial context, this characteristic modifies the critical growth of the nonlinearities in some senses. This fact allows us to study some well-known problems under new perspectives. For this purpose, we have considered three different classes of problems with critical nonlinearity which presents the weight of Hénon. Firstly, we have studied the class of problem with a Trudinger- Moser nonlinearity in critical range in R². In the subcritical case, there was no diference if we have looked for weak solutions in H¹₀ (B₁) or in H¹₀,rad(B₁). Nevertheless, in the critical case we have needed to adapt some hypotheses when we have changed the space where we were seeking the solutions. For the second problem, we have kept working with exponential nonlinearity in R², but we were treating an Ambrosseti-Prodi problem for which we have searched two weak solutions. In the subcritical case, analogously to first problem, the radially symmetric solutions were obtained as the solutions in H¹₀ (B₁), what have not happened in the critical case. Thus, again some assumptions have had to depend on the context where we were searching for the solutions. Lastly, we have studied a natural version of the second problem with the nonlinearity involving critical Sobolev growth in Rᴺ (N ≥ 3). In this last problem, we have searched the existence of solutions only in the radial critical case because the others cases were almost identical to problems with nonlinearities without the weight of Hénon.CNPqNeste trabalho, utilizando métodos variacionais investigamos a existência de soluções para algumas equações do tipo Hénon, que são caracterizadas pela presença do peso lxlᵅ na não-linearidade com α> 0. Quando estamos trabalhando no contexto radial, essa característica modifica o crescimento critico das não-linearidades em alguns sentidos. Este fato nos permite estudar problemas bem conhecidos sob novas perspectivas. Com este propósito, consideramos três classes diferentes de problemas com uma não-linearidade que apresenta o peso de Hénon. Em primeiro lugar, estudamos a classe de problema envolvendo uma não-linearidade do tipo Trudinger-Moser com imagem critica em R². No caso subcrítico, não houve diferença se procuramos soluções fracas em H¹₀ (B₁) ou em H¹₀,rad(B₁). No entanto, no caso crítico, precisamos adaptar algumas hipóteses quando mudamos o espaço onde buscávamos as soluções. Para o segundo problema, continuamos trabalhando com uma não-linearidade exponencial em R², mas desta vez tratando de um problema do tipo Ambrosseti-Prodi para o qual buscamos duas soluções fracas. No caso subcrítico, analogamente ao primeiro problema, as soluções radialmente simétricas foram obtidas do mesmo modo das soluções em H¹₀ (B₁), o que não aconteceu no caso crítico. Assim, algumas hipótese novamente tiveram que depender do contexto em que buscávamos as soluções. Por fim, estudamos uma versão natural do segundo problema com a não-linearidade envolvendo o crescimento crítico do tipo Sobolev em Rᴺ (N ≥ 3). Neste ultimo problema, pesquisamos a existência apenas de soluções radiais no caso crítico porque os outros casos eram quase idênticos a problemas com não-linearidades sem o peso de Hénon.engUniversidade 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 parciaisHénon type equations with nonlinearities in the critical growth rangeinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/doctoralThesisdoutoradoreponame:Repositório Institucional da UFPEinstname:Universidade Federal de Pernambuco (UFPE)instacron:UFPETHUMBNAILTESE Eudes Mendes Barboza.pdf.jpgTESE Eudes Mendes Barboza.pdf.jpgGenerated Thumbnailimage/jpeg1277https://repositorio.ufpe.br/bitstream/123456789/25399/6/TESE%20Eudes%20Mendes%20Barboza.pdf.jpg1fe21ea059e94472c62db032b1dc7245MD56ORIGINALTESE Eudes Mendes Barboza.pdfTESE Eudes Mendes Barboza.pdfapplication/pdf6171607https://repositorio.ufpe.br/bitstream/123456789/25399/1/TESE%20Eudes%20Mendes%20Barboza.pdf74a89b570c111f3b75e648044f5d5a80MD51LICENSElicense.txtlicense.txttext/plain; charset=utf-82311https://repositorio.ufpe.br/bitstream/123456789/25399/3/license.txt4b8a02c7f2818eaf00dcf2260dd5eb08MD53CC-LICENSElicense_rdflicense_rdfapplication/rdf+xml; charset=utf-8811https://repositorio.ufpe.br/bitstream/123456789/25399/4/license_rdfe39d27027a6cc9cb039ad269a5db8e34MD54TEXTTESE Eudes Mendes Barboza.pdf.txtTESE Eudes Mendes Barboza.pdf.txtExtracted texttext/plain210828https://repositorio.ufpe.br/bitstream/123456789/25399/5/TESE%20Eudes%20Mendes%20Barboza.pdf.txt21737bd2c78b3172ef4815c95bc46876MD55123456789/253992019-10-26 01:16:32.934oai:repositorio.ufpe.br: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Repositório InstitucionalPUBhttps://repositorio.ufpe.br/oai/requestattena@ufpe.bropendoar:22212019-10-26T04:16:32Repositório Institucional da UFPE - Universidade Federal de Pernambuco (UFPE)false |
dc.title.pt_BR.fl_str_mv |
Hénon type equations with nonlinearities in the critical growth range |
title |
Hénon type equations with nonlinearities in the critical growth range |
spellingShingle |
Hénon type equations with nonlinearities in the critical growth range BARBOZA, Eudes Mendes Matemática Análise não linear Equações diferenciais parciais |
title_short |
Hénon type equations with nonlinearities in the critical growth range |
title_full |
Hénon type equations with nonlinearities in the critical growth range |
title_fullStr |
Hénon type equations with nonlinearities in the critical growth range |
title_full_unstemmed |
Hénon type equations with nonlinearities in the critical growth range |
title_sort |
Hénon type equations with nonlinearities in the critical growth range |
author |
BARBOZA, Eudes Mendes |
author_facet |
BARBOZA, Eudes Mendes |
author_role |
author |
dc.contributor.authorLattes.pt_BR.fl_str_mv |
http://lattes.cnpq.br/9426464458648172 |
dc.contributor.advisorLattes.pt_BR.fl_str_mv |
http://lattes.cnpq.br/6069135199129029 |
dc.contributor.author.fl_str_mv |
BARBOZA, Eudes Mendes |
dc.contributor.advisor1.fl_str_mv |
DO Ó, Joao Marcos Bezerra |
contributor_str_mv |
DO Ó, Joao Marcos Bezerra |
dc.subject.por.fl_str_mv |
Matemática Análise não linear Equações diferenciais parciais |
topic |
Matemática Análise não linear Equações diferenciais parciais |
description |
In this work, using variational methods we have investigated the existence of solutions for some Hénon type equations, which are characterized by the presence of the weight lxlᵅ in the nonlinearity with α > 0. When we are working in the radial context, this characteristic modifies the critical growth of the nonlinearities in some senses. This fact allows us to study some well-known problems under new perspectives. For this purpose, we have considered three different classes of problems with critical nonlinearity which presents the weight of Hénon. Firstly, we have studied the class of problem with a Trudinger- Moser nonlinearity in critical range in R². In the subcritical case, there was no diference if we have looked for weak solutions in H¹₀ (B₁) or in H¹₀,rad(B₁). Nevertheless, in the critical case we have needed to adapt some hypotheses when we have changed the space where we were seeking the solutions. For the second problem, we have kept working with exponential nonlinearity in R², but we were treating an Ambrosseti-Prodi problem for which we have searched two weak solutions. In the subcritical case, analogously to first problem, the radially symmetric solutions were obtained as the solutions in H¹₀ (B₁), what have not happened in the critical case. Thus, again some assumptions have had to depend on the context where we were searching for the solutions. Lastly, we have studied a natural version of the second problem with the nonlinearity involving critical Sobolev growth in Rᴺ (N ≥ 3). In this last problem, we have searched the existence of solutions only in the radial critical case because the others cases were almost identical to problems with nonlinearities without the weight of Hénon. |
publishDate |
2017 |
dc.date.issued.fl_str_mv |
2017-05-30 |
dc.date.accessioned.fl_str_mv |
2018-08-06T17:23:24Z |
dc.date.available.fl_str_mv |
2018-08-06T17:23:24Z |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/doctoralThesis |
format |
doctoralThesis |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
https://repositorio.ufpe.br/handle/123456789/25399 |
url |
https://repositorio.ufpe.br/handle/123456789/25399 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
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 |
rights_invalid_str_mv |
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 |
publisher.none.fl_str_mv |
Universidade Federal de Pernambuco |
dc.source.none.fl_str_mv |
reponame:Repositório Institucional da UFPE instname:Universidade Federal de Pernambuco (UFPE) instacron:UFPE |
instname_str |
Universidade Federal de Pernambuco (UFPE) |
instacron_str |
UFPE |
institution |
UFPE |
reponame_str |
Repositório Institucional da UFPE |
collection |
Repositório Institucional da UFPE |
bitstream.url.fl_str_mv |
https://repositorio.ufpe.br/bitstream/123456789/25399/6/TESE%20Eudes%20Mendes%20Barboza.pdf.jpg https://repositorio.ufpe.br/bitstream/123456789/25399/1/TESE%20Eudes%20Mendes%20Barboza.pdf https://repositorio.ufpe.br/bitstream/123456789/25399/3/license.txt https://repositorio.ufpe.br/bitstream/123456789/25399/4/license_rdf https://repositorio.ufpe.br/bitstream/123456789/25399/5/TESE%20Eudes%20Mendes%20Barboza.pdf.txt |
bitstream.checksum.fl_str_mv |
1fe21ea059e94472c62db032b1dc7245 74a89b570c111f3b75e648044f5d5a80 4b8a02c7f2818eaf00dcf2260dd5eb08 e39d27027a6cc9cb039ad269a5db8e34 21737bd2c78b3172ef4815c95bc46876 |
bitstream.checksumAlgorithm.fl_str_mv |
MD5 MD5 MD5 MD5 MD5 |
repository.name.fl_str_mv |
Repositório Institucional da UFPE - Universidade Federal de Pernambuco (UFPE) |
repository.mail.fl_str_mv |
attena@ufpe.br |
_version_ |
1802310801105616896 |