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Some classes of elliptic problems with singular nonlinearities

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...

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Main Author:	CLEMENTE, Rodrigo Genuino
Other Authors:	do Ó, João Marcos Bezerra
Format:	doctoralThesis
Language:	por
Published:	Universidade Federal de Pernambuco 2018
Subjects:	Matemática Análise não-linear Equações diferenciais parciais
Online Access:	https://repositorio.ufpe.br/handle/123456789/23433
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spelling	ir-123456789-234332019-10-26T01:55:49Z Some classes of elliptic problems with singular nonlinearities CLEMENTE, Rodrigo Genuino do Ó, João Marcos Bezerra http://lattes.cnpq.br/4351609162717260 http://lattes.cnpq.br/6069135199129029 Matemática Análise não-linear Equações diferenciais parciais 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. CAPES Problemas 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. 2018-01-30T19:06:26Z 2018-01-30T19:06:26Z 2016-02-24 doctoralThesis https://repositorio.ufpe.br/handle/123456789/23433 por Attribution-NonCommercial-NoDerivs 3.0 Brazil http://creativecommons.org/licenses/by-nc-nd/3.0/br/ application/pdf Universidade Federal de Pernambuco UFPE Brasil Programa de Pos Graduacao em Matematica
institution	REPOSITORIO UFPE
collection	REPOSITORIO UFPE
language	por
topic	Matemática Análise não-linear Equações diferenciais parciais
spellingShingle	Matemática Análise não-linear Equações diferenciais parciais CLEMENTE, Rodrigo Genuino Some classes of elliptic problems with singular nonlinearities
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.
author2	do Ó, João Marcos Bezerra
format	doctoralThesis
author	CLEMENTE, Rodrigo Genuino
author_sort	CLEMENTE, Rodrigo Genuino
title	Some classes of elliptic problems with singular nonlinearities
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
publisher	Universidade Federal de Pernambuco
publishDate	2018
url	https://repositorio.ufpe.br/handle/123456789/23433
_version_	1648655424124092416
score	13.657419

Some classes of elliptic problems with singular nonlinearities

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