# Equations aux dérivées partielles elliptiques du quatrième ordre avec exposants critiques de Sobolev sur les variétés riemanniennes avec et sans bord

Abstract : The subject of this thesis consists in the study, on compact riemmannian manifold $(V_n,g)$ of dimension $n>4$, of elliptic partial differential equation $(E)\; \Delta^2u+\nabla [a(x)\nabla u] +h(x)u= f(x)|u|^(N-2)u$ in which $a$,$h$,$f$ are $C^\infty$ fonctions with $f(x)$ constant fonction or everywhere strictly positive and $N=(2n\over((n-4)))$ is the critical exponent.Using the variational method, we prove on the main theorem that the equation (E) has non zero solution $C^((5,\alpha))(V)$ $0<\alpha<1$ if a certain condition which depends on the best constant in the Sobolev inequality ($H_2\subset L_(2n\over(n-4))$) is respected. Moreover, we prove that if $a$ and $h$ are accurate constant fonctions, the solution of the equation is positive and $C^\infty$. When $n\geq 6$, we present some applications of main theorem. In the last part we work on a compact riemannian manifold with boundary of dimension n, $(\overline(W)_n,g )$, we study the problem $(P_N) \; \left\lbrace \begin(array)(c) \Delta^2 v+\nabla [a(x)\nabla u] +h(x) v= f(x)|v |^(N-2)v \; \hbox(on)\; W \\ \Delta v =\delta \, , \, v = \eta \;\hbox(on) \;\partial W \end(array)\right.$ in which $\delta$, $\eta$ and $f$ are $C^\infty (\overline (W))$ fonctions with $f(x)$ everywhere strictly positive and we prove the existence of the solution for the problem $P_N$.
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Theses
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https://tel.archives-ouvertes.fr/tel-00003179
Contributor : Daniela Caraffa Bernard <>
Submitted on : Saturday, July 26, 2003 - 2:22:29 PM
Last modification on : Wednesday, December 9, 2020 - 3:05:25 PM
Long-term archiving on: : Friday, April 2, 2010 - 7:22:45 PM

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• HAL Id : tel-00003179, version 1

### Citation

Daniela Caraffa Bernard. Equations aux dérivées partielles elliptiques du quatrième ordre avec exposants critiques de Sobolev sur les variétés riemanniennes avec et sans bord. Mathématiques [math]. Université Pierre et Marie Curie - Paris VI, 2003. Français. ⟨tel-00003179⟩

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