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Teoremas Tipo Liouville e Desigualdades Tipo Harnack para Equações Elípticas Semilineares via Método Moving SpheresLima, Jalman Alves de 10 June 2011 (has links)
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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.
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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 indexMtiri, 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)
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Fully linear elliptic equations and semilinear fractionnal elliptic equationsChen, 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 ...
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