In this dissertation we study positive solutions to a singular p-Laplacian elliptic boundary value problem on a bounded domain with smooth boundary when a positive parameter varies. Our main focus is the analysis of a challenging class of singular p-Laplacian problems. We establish the existence of a positive solution for all positive values of the parameter and the existence of at least two positive solutions for a certain explicit range of the parameter. In the Laplacian case, we also prove the uniqueness of the positive solution for large values of the parameter. We extend our existence and multiplicity results to classes of singular systems and to the case when a domain is an exterior domain. We prove our existence and multiplicity results by the method of sub and supersolutions and our uniqueness result by establishing apriori and boundary estimates. Such results are well known in the literature for the nonsingular case. In this study, we extend these results to the more difficult singular case.

回顧一個重要的非線性二階方程式 d/dt(t^p(du/dt))+(-)t^(sigma)u^n=0, 這個方程式有許多有趣的物理應用，以Emden方程式的形式發生在天體物理學中；也以Fermi-Thomas方程式的形式出現在原子物理內。對於此類型的非線性方程式可以用來更頻繁且深入的探討數學物理，雖然目前仍存在著些許不確定性，不過如果在未來能有更全面的了解，這將有助於用來決定物理解的性質。 在這篇論文當中，我們討論微分方程式 t^2u"=u^p，p屬於N-{1}, 其正解的性質。這個方程式是著名的 Emden-Fowler 方程式的一種特殊情形， 我們可以得到其解的一些有趣的現象及結果。 / Recall the important nonlinear second-order equation d/dt(t^p(du/dt))+(-)t^(sigma)u^n=0, this equation has several interesting physical applications, occurring in astrophysics in the form of the Emden equation and in atomic physics in the form of the Fermi-Thomas equation. These seems a little doubt that nonlinear equations of this type would enter with greater frequency into mathematical physics, were it more widely known with what ease the properties of the physical solutions can be determined. In this paper we discuss the property of positive solution of the ordinary differential equation t^2u"=u^p, p belongs to N-{1}, this equation is a special case of the well-known Emden-Fowler equation, we obtain some interesting phenomena and resulits for solutions.

Submitted by ANA KARLA PEREIRA RODRIGUES (anakarla_@hotmail.com) on 2017-08-15T12:49:10Z No. of bitstreams: 1 arquivototal.pdf: 1323834 bytes, checksum: 530efbd6b56f11c5cc1b4369c8c44888 (MD5) / Made available in DSpace on 2017-08-15T12:49:10Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 1323834 bytes, checksum: 530efbd6b56f11c5cc1b4369c8c44888 (MD5) Previous issue date: 2016-02-29 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work we establish existence, multiplicity and concentration of positive solutions for the following class of problem 8<: 􀀀div􀀀 2 ( jruj)ru + V (x) (juj)u = f(u); in RN; u 2 W1; (RN); u > 0 in RN; where N 2, is a positive parameter, ; V; f are functions satisfying technical conditions that will be presented throughout the thesis and (t) = Rjtj 0 (s)sds. The main tools used are Variational methods, Lusternik-Schnirelman of category, Penalization methods and properties of Orlicz-Sobolev spaces. / Neste trabalho estabelecemos resultados de existência, multiplicidade e concentração de soluções positivas para a seguinte classe de problemas quasilineares 8<: 􀀀div􀀀 2 ( jruj)ru + V (x) (juj)u = f(u); em RN; u 2 W1; (RN); u > 0 em RN; onde N 2, é um parâmetro positivo, ; V; f são funções satisfazendo condições técnicas que serão apresentadas ao longo da tese e (t) = Rjtj 0 (s)sds. As principais ferramentas utilizadas são os Métodos Variacionais, Categoria de Lusternik-Schnirelman, Método de Penalização e propriedades dos espaços de Orlicz-Sobolev.

Made available in DSpace on 2016-06-02T20:27:36Z (GMT). No. of bitstreams: 1 1610.pdf: 953018 bytes, checksum: 71de779ec49ee3cef03c3060c45a97f3 (MD5) Previous issue date: 2007-11-20 / Financiadora de Estudos e Projetos / In this work, we will study the existence and nonexistence of positive weak solutions for two classes of elliptic systems with weights. The first class will involve nonlinearities of the type positone and semipositone. We will prove a strong maximum principle, and we will obtain some properties of the first eigenfunction of the eigenvalue problem associated to our operator, and also we will prove the sub and supersolution method. The second class will involve a nonlinear perturbation. We will use the variational methods to study the subcritical and critical situations, and under certain hypotheses, we will show the existence of a second weak solution. / Neste trabalho, estudaremos a existência e inexistência de solução fraca positiva para duas classes de sistemas elípticos com pesos. A primeira classe envolverá não linearidades do tipo positônico e semipositônico. Provaremos um princípio de máximo forte, e obteremos algumas propriedades da primeira autofunção do problema de autovalor associado ao nosso operador, e também provaremos o método de sub e supersolução. A segunda classe que consideraremos terá uma perturbação não linear. Usaremos os métodos variacionais para estudar tanto a situação subcrítica quanto à situação crítica, e sob certas hipóteses, mostraremos a existência de uma segunda solução fraca.