Dissertations, Theses, and Capstone Projects

Date of Degree

2-2018

Document Type

Dissertation

Degree Name

Ph.D.

Program

Mathematics

Advisor

Marcello Lucia

Committee Members

Zheng Huang

Leon Karp

Subject Categories

Analysis

Keywords

Polyharmonic Dirichlet problems, perturbation from symmetry, critical point theory, variational methods

Abstract

In this dissertation we prove new results on the existence of infinitely many solutions to nonlinear partial differential equations that are perturbed from symmetry. Our main theorems focus on polyharmonic Dirichlet problems with exponential nonlinearities, and are now published in Topol. Methods Nonlinear Anal. Vol. 50, No.1, (2017), 27-63. In chapter 1 we give an introduction to the problem, its history, and the perturbation argument itself. In chapter 2 we prove the variational principle of Bolle on the behavior of critical values under perturbation, and the variational principle of Tanaka on the existence of critical points of large augmented Morse index. In chapter 3 we use the framework created by Birman and Solomyak for deriving eigenvalue estimates to find alternatives of the CLR inequality specifically designed for our particular nonlinear PDE applications. Chapters 2 and 3 comprise the tools of the perturbation argument. In chapter 4 we bring everything together and prove our main results. We also include new results on non-homogeneous boundary values, and unbounded domains.

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