Neste trabalho, estudamos a existˆencia e multiplicidade de solu¸c˜oes de certos problemas p-sublineares envolvendo o operador p-laplaciano usando teoria de Morse. / The purpose of this text is to provide a didactic exposition of the paper “Solutions of p-sublinear p-Laplacian equation via Morse theory” by Yuxia Guo and Jiaquan Liu [8]. This paper addresses the existence and multiplicity of solutions for the problem where is a smooth, bounded domain of RN, p is the p-Laplacian operator and f satisfies certain conditions, in particular f is p-sublinear at 0. Morse theory is used to infer the existence of critical points of a functional associated to this problem. In Chapter 2, we introduce the necessary Morse theoretic concepts, assuming basic knowledge of singular homology theory. In Chapter 3, we introduce basic properties of the p-Laplacian operator, assuming knowledge of Sobolev spaces, including imbedding and compactness results. Finally, in Chapter 4, we follow Guo and Liu’s paper itself.

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In this thesis, we consider the following one dimensional p-Laplacian eigenvalue problem: -((y¡¦/s)^(p-1))¡¦+(p-1)(q-£fw)y^(p-1)=0 a.e. on (0,1) (0.1) and satisfy £\y(0)+ £\ ¡¦ (y¡¦(0)/s(0))=0 £]y(1)+£]¡¦ (y¡¦(1)/s(1))=0 (0.2) where f^(p-1)=|f|^p-2 f=|f|^p-1 sgnf; £\, £\¡¦, £], £]¡¦ ∈R such that £\^2+£\¡¦^2>0 and£]^2+£]¡¦^2>0; and the functions s,q,w are required to satisfy (1) s,q,w∈L^1(0,1); (2) for 0≤x≤1, we have s≥0,w≥0 a.e.; (3) for any x∈ (0,1), ¡ì_0^1 s(t)dt>0, ¡ì_0^x w(t)dt>0,and¡ì_x^1 w(t)dt>0; (4) if for some x_1<x_2,we have¡ì_ x1^x2 w(t)dt=0,then¡ì_ x1^x2 |q(t)|dt=0; (5) for all n∈N, there is a partition {£a_i^(n)}_i=1 ^2n of [0,1] such that for any 0<k≤n-1, ¡ì_£a_2k^(n)^ £a_2k+1^(n) w>0 and ¡ì_£a_2k+1^(n)^ £a_2k+2^(n) s>0. We call the above conditions Atkinson conditions, first introduce in [1].There conditions include the case when s,q,w∈L^1(0,1) and s,w>0 a.e. We use a generalized Prufer substitution and Caratheodory theorem to prove the existence and uniqueness for the solution of the initial value problem of (0.1) above. Then we generalize the Sturm oscillation theorem to one dimensional p-Laplacian and establish the Sturm-Liouville properties of the p-Laplacian operators with L^1 coefficient functions. Our results filled up some gaps in Binding-Drabek [3].

Neste trabalho, estudamos a existˆencia e multiplicidade de solu¸c˜oes de certos problemas p-sublineares envolvendo o operador p-laplaciano usando teoria de Morse. / The purpose of this text is to provide a didactic exposition of the paper “Solutions of p-sublinear p-Laplacian equation via Morse theory” by Yuxia Guo and Jiaquan Liu [8]. This paper addresses the existence and multiplicity of solutions for the problem where is a smooth, bounded domain of RN, p is the p-Laplacian operator and f satisfies certain conditions, in particular f is p-sublinear at 0. Morse theory is used to infer the existence of critical points of a functional associated to this problem. In Chapter 2, we introduce the necessary Morse theoretic concepts, assuming basic knowledge of singular homology theory. In Chapter 3, we introduce basic properties of the p-Laplacian operator, assuming knowledge of Sobolev spaces, including imbedding and compactness results. Finally, in Chapter 4, we follow Guo and Liu’s paper itself.

Neste trabalho, estudamos a existˆencia e multiplicidade de solu¸c˜oes de certos problemas p-sublineares envolvendo o operador p-laplaciano usando teoria de Morse. / The purpose of this text is to provide a didactic exposition of the paper “Solutions of p-sublinear p-Laplacian equation via Morse theory” by Yuxia Guo and Jiaquan Liu [8]. This paper addresses the existence and multiplicity of solutions for the problem where is a smooth, bounded domain of RN, p is the p-Laplacian operator and f satisfies certain conditions, in particular f is p-sublinear at 0. Morse theory is used to infer the existence of critical points of a functional associated to this problem. In Chapter 2, we introduce the necessary Morse theoretic concepts, assuming basic knowledge of singular homology theory. In Chapter 3, we introduce basic properties of the p-Laplacian operator, assuming knowledge of Sobolev spaces, including imbedding and compactness results. Finally, in Chapter 4, we follow Guo and Liu’s paper itself.

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.

We characterize trees that have greatest maximum p-Laplacian eigenvalue among all trees with a given degree sequence. We show that such extremal trees can be obtained by breadth-first search where the vertex degrees are non-increasing. These trees are uniquely determined up to isomorphism. Moreover, their structure does not depend on p. / Series: Research Report Series / Department of Statistics and Mathematics

In this theses, an iterativ linear solver for the non-linear p-Poisson problem is introduced. After the theoretical convergence results some numerical examples of a fully adaptive solver are presented.

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.

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