Award Date

5-1-2022

Degree Type

Thesis

Degree Name

Master of Science (MS)

Department

Mathematical Sciences

First Committee Member

David Costa

Second Committee Member

Le Chen

Third Committee Member

Zhijian Wu

Fourth Committee Member

Amei Amei

Fifth Committee Member

Zhaohuan Zhu

Number of Pages

55

Abstract

Nonlinear elliptic partial differential equations on bounded domains arise in several different areas of mathematics that include geometry, mathematical physics, and the calculus of variations. The Br ́ezis-Nirenberg problem is concerned with a boundary-value problem that is intimately connected to the existence of positive solutions of the Yamabe problem, of non-minimal solutions to Yang-Mills functionals, and of extremal functions to several important inequalities. Results on existence and uniqueness have been obtained in cases when the exponent is sub-critical, but such results have not been obtained when the exponent is critical due to a lack of compactness. The earliest results obtained in this direction were obtained by Br ́ezis and Nirenberg. The goal of this thesis is to serve as a survey of the various results regarding this variational problem.

Keywords

Critical Exponent; Differential Geometry; Existence and Multiplicity; Nonlinear PDEs; Topological Methods; Variational Methods

Disciplines

Applied Mathematics | Mathematics

File Format

pdf

File Size

598 KB

Degree Grantor

University of Nevada, Las Vegas

Language

English

Rights

IN COPYRIGHT. For more information about this rights statement, please visit http://rightsstatements.org/vocab/InC/1.0/

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