从罗尔定理到山路定理

From Rolle’s Theorem to Mountain Pass Theorem

摘要: 山路定理是现代临界点理论的标志性结果，它是证明非线性椭圆型偏微分方程有解的重要工具。本文旨在研究有限维山路定理的初等证明以及有限维山路定理与微积分之间的关系。再在此基础上，给出山路定理的应用实例。本文从教学实际出发，为学生进入研究生阶段理解非线性分析中的重要定理提供有益帮助。

Abstract: The mountain pass theorem is a landmark result of modern critical point theory, which is an important tool for proving the existence of solutions to nonlinear elliptic partial differential equations. This article aims to study the elementary proof of the finite-dimensional mountain path theorem and the relationship between the finite-dimensional mountain pass theorem and calculus. On this basis, an application example of the mountain pass theorem is given. This article starts from the actual teaching situation, providing beneficial help for students to understand important theorems in nonlinear analysis when they enter the graduate stage.