My thesis work started in the summer of 2005 as a three way joint project by Professor Castro and Mr. John Kwon and myself. A paper from this joint project was written and the content now forms my thesis.

In this paper we prove that div(|x|β∇u)+|x|αf(u)=0, inB u = 0 on ∂B has infinitely many solutions when f is superlinear and grows subcritically for u ≥ 0 and up to critically for u less than 0 with 10, 13 N+β−2 N+β−2 We make extensive use of Pohozaev identities and phase plane and energy arguments.

In this dissertation, we study the existence and nonexistence of positive radial solutions for classes of quasilinear elliptic equations and systems in a ball with Dirichlet boundary conditions. Our nonlinearities are asymptotically p-linear at infinity and are allowed to be singular at zero with non-positone structure, which have not been considered in the literature. In the one parameter single equation problem, we are able to show the existence of a positive radial solution with precise lower bound estimate for a certain range of the parameter. We also extend the study to a class of asymptotically p-linear system with two parameters and in the presence of singularities. We establish the existence of a positive solution with a precise lower bound estimate when the product of the parameters is in a certain range. Necessary and sufficient conditions for the existence of a positive solution are also obtained for both the single equation and system under additional assumptions. Our approach is based on the Schauder Fixed Point Theorem.

We establish the existence of radial solutions to the p-Laplacian equation ∆p u + f(u)=0 in RN, where f behaves like |u|q-1 u when u is large and f(u) < 0 for small positive u. We show that for each nonnegative integer n, there is a localized solution u which has exactly n zeros. Also, we look for radial solutions of a superlinear Dirichlet problem in a ball. We show that for each nonnegative integer n, there is a solution u which has exactly n zeros. Here we give an alternate proof to that which was given by Castro and Kurepa.

In this dissertation, we study the existence and nonexistence of positive solutions to some classes of singular p-Laplacian boundary value problems with a parameter. In the first study, we discuss positive solutions for a class of sublinear Dirichlet p- Laplacian equations and systems with sign-changing coefficients on a bounded domain of Rn via Schauder Fixed Point Theorem and the method of sub- and supersolutions. Under certain conditions, we show the existence of positive solutions when the parameter is large and nonexistence when the parameter is small. In the second study, we discuss positive radial solutions for a class of superlinear p- Laplacian problems with nonlinear boundary conditions on an exterior domain via degree theory and fixed point approach. Under certain conditions, we show the existence of positive solutions when the paprameter is small and nonexistence when the paramter is large. Our results provide extensions of corresponding ones in the literature from the Laplacian to the p-Laplacian, and can be applied to the challenging infinite semipositone case

The thesis presents the results of our research on symmetry for some semilinear elliptic problems and on existence of solution for quasilinear problems involving singularities. The text is composed of two parts, each of which begins with a specific introduction. The first part is devoted to symmetry and symmetry-breaking results. We study a class of partial differential equations involving radial weights on balls, annuli or $R^N$ --where these weights are unbounded--. We show in particular that on unbounded domains, focusing on symmetric functions permits to recover compactness, which implies existence of solutions. Then, we stress the fact that symmetry-breaking occurs on bounded domains, depending both on the weights and on the nonlinearity of the equation. We also show that for the considered class of problems, the multibumps-solution phenomenon appears on the annulus as well as on the ball. The second part of the thesis is devoted to partial and ordinary differential equations with singularities. Using concentration-compactness tools, we show that a rather large class of functionals is lower semi-continuous, leading to the existence of a ground state solution. We also focus on the unicity of solutions for such a class of problems.

Submitted by Johnny Rodrigues (johnnyrodrigues@ufcg.edu.br) on 2018-07-24T16:13:25Z No. of bitstreams: 1 JACKSON JONAS SILVA COSTA - DISSERTAÇÃO PPGMAT 2010..pdf: 1326944 bytes, checksum: 5f5bb0430a12c8bdeabdea73796d8578 (MD5) / Made available in DSpace on 2018-07-24T16:13:25Z (GMT). No. of bitstreams: 1 JACKSON JONAS SILVA COSTA - DISSERTAÇÃO PPGMAT 2010..pdf: 1326944 bytes, checksum: 5f5bb0430a12c8bdeabdea73796d8578 (MD5) Previous issue date: 2010-04 / Capes / Para visualizar o resumo recomendamos do download do arquivo uma vez que o mesmo utiliza fórmulas ou equações matemáticas que não puderam ser transcritas neste espaço. / In order to view the summary we recommend downloading the file as it uses mathematical formulas or equations that could not be transcribed in this space.