Thesis (Ph.D.)--Boston University / PLEASE NOTE: Boston University Libraries did not receive an Authorization To Manage form for this thesis or dissertation. It is therefore not openly accessible, though it may be available by request. If you are the author or principal advisor of this work and would like to request open access for it, please contact us at open-help@bu.edu. Thank you. / The first boundary value problem for eΔu + a(x,y,e)ux + b(x,y,e)uy +c(x,y,e)u = d(x,y,e) for small e. This problem with coefficients independent of e was treated by Norman Levinson and appeared in Annals of Mathematics, Vol. 51, No. 2, March, 1950 [TRUNCATED]. / 2999-01-01

We are concerned with the so-called asymptotic Plateau problem in hyperbolic space. That is, to prove the existence of hypersurfaces in hyperbolic space whose principal curvatures satisfy a general curvature relation and has a precribed asymptotic boundary at infinity. In this thesis, by following the method of Bo Guan, Joel Spruck and their collaborators, we solve the problem with the aid of an additional assumption. In particular, our result applies to hypersurfaces whose principal curvatures lie in the k-th Garding cone and has constant (k,k-1) curvature quotient. / Thesis / Master of Science (MSc)

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Dans cette thèse, nous donnons des résultats d’existence, de non-existence, d’unicité et de multiplicité de solutions pour des équations aux dérivées partielles avec croissance critique dans le gradient. Les principales méthodes utilisées dans nos preuves sont des arguments variationnels, la théorie des sous et sur-solutions, des estimations à priori et la théorie de la bifurcation. La thèse se compose de six chapitres. Dans le chapitre 0 nous introduisons le sujet de thèse et nous présentons les résultats principaux. Le chapitre 1 porte sur l’´étude d’une équation du type p-Laplacien avec croissance critique dans le gradient et dépendant d’un paramètre. En fonction de l’intervalle où se trouve le paramètre, nous obtenons l’existence et l’unicité d’une solution ou nous montrons l’existence et la multiplicité de solutions. Dans les chapitres 2 et 3, nous poursuivons notre étude dans le cas où l’opérateur utilisé est le Laplacien mais, contrairement au chapitre 1, nous étudions le cas où les coefficients changent de signe. Nous obtenons à nouveau des résultats d’existence et de multiplicité de solutions. Dans le chapitre 4, nous étudions des problèmes nonlocaux du type Laplacien fractionnaire avec différents termes de gradient non-local. Nous montrons des résultats d’existence et de non-existence de solutions pour différentes équations de ce type. Finalement, dans le chapitre 5 nous présentons quelques problèmes ouverts liés au contenu de la thèse et des perspectives de recherche. / In this thesis, we provide existence, non-existence, uniqueness and multiplicity results for partial differential equations with critical growth in the gradient. The principal techniques employed in our proofs are variational techniques, lower and upper solution theory, a priori estimates and bifurcation theory. The thesis consists of six chapters. In chapter 0, we introduce the topic of the thesis and we present the main results. Chapter 1 deals with a p-Laplacian type equation with critical growth in the gradient. This equation will depend on a real parameter. Depending on the interval where this parameter lives, we obtain the existence and uniqueness of one solution or we prove the existence and multiplicity of solutions. In chapters 2 and 3, we continue our study in the case where the operator is the Laplacian. However, unlike chapter 1, we study the case where the coefficient functions may change sign. We obtain again existence and multiplicity results. In chapter 4, we study non-local problems of fractional Laplacian type with different non-local gradient terms. We prove existence and non-existence results for different equations of this type. Finally, in chapter 5, we present some open problems related to the content of the thesis and some research perspectives.