An Exposition on Critical Point Theory with Applications

Abstract

Subham De

Critical Point Theory plays a pivotal role in the study of Partial Differential Equations (PDEs), particularly in investigating the existence, uniqueness, and multiplicity of weak solutions to elliptic PDEs under specified boundary conditions. This article offers a concise survey of key concepts, including differentiation on Banach spaces, the analysis of maxima and minima, and their applications to PDEs. Additionally, we explore the construction of weak topologies on Banach spaces before introducing the Variational Principle and its relevance to solving PDEs. In the final section, we focus on results pertaining to the existence of weak solutions for Dirichlet boundary value problems under specific conditions. A highlight of this work is the proof of Rabinowitz’s Saddle Point Theorem via the Brouwer Degree method. Researchers interested in exploring the themes covered in this paper will find the reference section to be a valuable resource for further study.

PDF