The Mountain Pass Theorem and Applications

Abstract

This project lies at the interface between Nonlinear Functional Anal- ysis, unconstrained Optimization and Critical point theory. It concerns mainly the Ambrosetti-Rabinowitz's Mountain Pass Theorem which is a min-max theorem at the heart of deep mathematics and plays a crucial role in solving many variational problems. As application, a model of Lane-Emden equation is considered. Minmax theorems characterize a critical value c of a functonal de ned on a Banach spaces as minmax over a suitable class subsets of X, A f of that is : c = inf sup f (x) . A∈S x∈A Variational methods refer to proofs established by showing that a suitable auxilliary function attains a minimum or has a critical point (see below). Minimum Variational principle can be viewed as a mathematical form of the principle of least action in Physics and justi es why so many results in Mathematics are related to varia- tional techniques since they have their origin in the physical sciences. The application of the Mountain Pass Theorem and more gen- erally those of Variational Techniques cover numerous theoretic as well as applied areas of mathematical sciences such as Partial Dif- ferential Equations, Optimization, Banach space geometry, Control theory, Economics and Game theory.