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1	Teoremas Tipo Liouville e Desigualdades Tipo Harnack para Equações Elípticas Semilineares via Método Moving Spheres Lima, Jalman Alves de 10 June 2011 (has links) Made available in DSpace on 2015-05-15T11:46:10Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 440995 bytes, checksum: d194a6a60d04b251160ec2e62f106e77 (MD5) Previous issue date: 2011-06-10 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work, we will do some applications of the Moving Spheres method, a variant of the method of Moving Planes, in order to obtain some Liouville-type theorems and some Harnack-type inequalities in Rn, as well as in the Euclidian half space Rn +. Our study focuses on, mostly, in the article written by Yan Yan Li and Lei Zhan [32], as well as some references of the same article. We concentrate in studying some properties of positive solutions of some semilinear elliptic partial differential equations with critical exponent and giving different proofs, improvements, and extensions of some previously established Liouville-type theorems and Harnack-type inequalities. / Neste trabalho, faremos algumas aplicações do método Moving Spheres, uma variante do método Moving Planes, na obtenção de alguns teoremas tipo Liouville e de algumas desigualdades tipo Harnack em Rn, bem como no semi-espaço euclidiano Rn +. Nosso estudo se concentra, marjoritariamente, no artigo do Yan Yan Li e Lei Zhang [32], bem como algumas referências do mesmo. Nos concentramos em estudar propriedades de soluções positivas de algumas equações diferenciais parciais elípticas semilineares com expoente crítico e dar provas diversificadas, refinamentos e extensões de alguns Teoremas tipo Liouville e desigualdades tipo Harnack já estabelecidos. Teorema tipo Liouville Desigualdades tipo Harnack Método Moving Spheres Liouville type theorem Harnack type inequalities Moving Spheres method CIENCIAS EXATAS E DA TERRA::MATEMATICA
2	Études des solutions de quelques équations aux dérivées partielles non linéaires via l'indice de Morse / Study of solutions of some nonlinear partial differential equations via the Morse index Mtiri, Foued 25 November 2016 (has links) Cette thèse porte principalement sur l'étude des solutions de certaines équations aux dérivées partielles elliptiques via l'indice de Morse, y compris des solutions stables, i.e. quand l'indice de Morse est égal à zéro. Elle comporte deux parties indépendantes.Dans la première partie, sous des hypothèses sur-linéaires et sous-critiques sur f, on établit d'abord une estimation explicite de la norme L [infini] des solutions de -Δu = f(u) avec u = 0 sur le bord, via leurs indices de Morse. On propose une approche plus transparente et plus souple que le travail de Yang [1998], ce qui nous permet de traiter des non linéarités très proches de la croissance critique. Les résultats obtenus nous ont motivé de travailler sur des équations polyharmoniques (-Δ)ku = f(x; u) avec notamment k = 2 et 3. Avec des hypothèses semblables à Yang [1998] sur f et des conditions au bord convenables, on obtient pour la première fois des estimations explicites de solution des équations polyhamoniques, via l'indice de Morse. Dans la seconde partie, on considère un système de Lane-Emden-Δu = ρ(x)vp; -Δv = ρ(x)u θ ; u; v > 0; dans RN; avec 1 < p< θ et un poids radial ρ strictement positif. Nous montrons la non-existence de solution stable en petites dimensions N. Nos résultats améliorent les travaux précédents de Cowan & Fazly [2012]; Fazly [2012]; Hu [2015], et fournissent notamment des résultats du type Liouville pour solution stable, en petites dimensions N, valables pour tout 1 < ρ min(4 3 ; θ) / The main concern of this thesis deals with the study of solutions of several elliptic partial differential equations via the Morse index, including the stable solutions, i.e. when the Morse index is zero. The thesis has two independent parts. In the first part, under suplinear and subcritical assumptions on f, we establish firstly some explicit estimation for the L1 norms of solutions to -Δu = f(u) avec u = 0 on the boundary, via its Morse index. We propose an approach more transparent and easier than the work of Yang [1998], which allow us to treat some nonlinearities very close to the critical growth. These results motivated us to consider the polyharmonic equations (-Δ)ku = f(x; u) with especially k = 2 and 3. With the hypothesis on f similar to Yang [1998] and appropriate boundary conditions, we obtain for the _rst time some explicit estimations of solution via its Morse index, for the polyharmonic equations.In the second part, we consider a Lane-Emden system -Δu = ρ(x)vp; -Δv = ρ(x)u_; u; v > 0; in RN; with 1 < p< θ and a radial positive weight ρ. We prove the non-existence of stable solution in small dimension case. Our results improve the previous works Cowan & Fazly [2012]; Fazly [2012]; Hu [2015], especially we prove some general Liouville type results for stable solutions in small dimension which hold true for any 1 < ρ min(4 3 ; θ) Indice de Morse Estimation elliptique Identité de Pohozaev Équations polyharmoniques Système de Lane-Emden avec poids Théorème du type Liouville Morse index Pohozaev identity Polyharmonic equation Weighted Lane-Emden system Stable solution Liouville type theorem Elliptic estimate 515.353
3	Fully linear elliptic equations and semilinear fractionnal elliptic equations Chen, Huyuan 10 January 2014 (has links) Cette thèse est divisée en six parties. La première partie est consacrée à l'étude de propriétés de Hadamard et à l'obtention de théorèmes de Liouville pour des solutions de viscosité d'équations aux dérivées partielles elliptiques complètement non-linéaires avec des termes de gradient, ... / This thesis is divided into six parts. The first part is devoted to prove Hadamard properties and Liouville type theorems for viscosity solutions of fully nonlinear elliptic partial differential equations with gradient term ... Propriété d'Hadamard Théorème de typr Liouville Solutions de ciscosité Laplacien fractionnaire Existence Unicité Comportement asymptotique Grandes solutions Mesures de Radon Mesure de Dirac Noyau de Green Capacité de Bessel Singularités isolées Solutions faibles Singularité faible Singularité forte Hadamard property Liouville type theorem Viscosity solution Fully nonlinear elliptic PDE Fractional Laplacian Existence Uniqueness Asymptotic behavior Blow-up solution Radon measure Dirac mass Green kernel Bessel capacities Isolated singularity Weak solution Weak singular solution Strong singular solution Propiedad de Hadamard Teorema del tipo de Liouville Soluciones Viscosas EDP elípticas completamente no lineales Laplaciano fraccionario Existencia, Unicidad Comportamiento asimptótico Soluciones blow-up Medida de Radon Masa de Dirac Núcleo de Green Capacidades de Bessel Singularidad aislada Soluciones débiles Soluciones singulares débiles Soluciones singulares fuertes

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