preprints.org > computer science and mathematics > analysis > doi: 10.20944/preprints202405.1021.v1

Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

Version 1 : Received: 14 May 2024 / Approved: 15 May 2024 / Online: 16 May 2024 (08:15:34 CEST)

The notion of Critical Point Theory has extensive application in the filed of Partial Differential Equations, one of which is studying the existence and commenting on the uniqueness and multiplicity (in case when the solution is not unique) of weak solutions to Elliptic PDEs under certain boundary conditions.\par This article provides a brief survey about specific concepts like : differentiation on Banach Spaces, maxima and minima and its applications to PDEs. Furthermore, we've discussed the notion of defining a weak topology on Banach Spaces, before introducing the Variational Principle and its applications.\par The epilogue of the article primarily deals with results which studies the existence of weak solutions to Dirichlet Boundary Value Problems under specific conditions. The most notable aspect of this article is the proof of \textit{Rabinowitz's Saddle Point Theorem} using the method of \textit{Brouwer Degree}. Readers who are highly motivated in pursuing research in any of the topics relevent to the contents of this paper will surely find the \texttt{References} section to be extremely resourceful.

critical point; saddle point; banach spaces; maxima & minima; weak convergence; constrained minimization; lagrange multiplier; sobolev embedding theorem; rellich’s theorem; palais-smale condition; pohozaev’s identity; variational principle; weak solution; critical exponent; brezis theorem; mountain pass theorem; rabinowitz saddle point theorem; brouwer degree

Computer Science and Mathematics, Analysis

* All users must log in before leaving a comment