A Variant of the Mountain Pass Theorem and Variational Gluing

Piero Montecchiari,Paul H Rabinowitz

doi:10.1007/s00032-020-00318-3

Piero Montecchiari, Paul H Rabinowitz

Open Access

https://doi.org/10.1007/s00032-020-00318-3

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Abstract

This paper surveys some recent work on a variant of the MountainPass Theorem that is applicable to some classes of differential equations involvingunbounded spatial or temporal domains. In particular its application to a systemof semilinear elliptic PDEs on R^n and to a family of Hamiltonian systems involvingdouble well potentials will also be discussed.

Highlights

The Mountain Pass Theorem is a useful tool for obtaining the existence of solutions of differential equations that arise as critical points of a functional, I, defined on a Banach space, E, or a subset thereof
This Proposition says along a subsequence, a (PS) sequence converges to a formal chain of widely spaced translates of solutions of (PDE), each of which corresponds to a critical point of I that is less than or equal to b
If Q satisfies one of the constraints defining A2 with equality, a comparison argument based on cutting and pasting arguments and a key inequality stemming from the splitting properties of S(a−, a+) and S(a+, a−)- see (4.3) and Proposition 4.4 of Section 4 - leads to a contradiction

Summary

Introduction

The Mountain Pass Theorem is a useful tool for obtaining the existence of solutions of differential equations that arise as critical points of a functional, I, defined on a Banach space, E, or a subset thereof. Our goal in this paper is to survey some recent work on a variant of the Mountain Pass Theorem that can be used to obtain the existence of basic mountain pass solutions in several settings such as the first two examples above and to indicate how the variational methods can be further employed to “glue” these solutions to construct multibump and multitransition solutions of I = 0 such as mentioned above For such constructions as well as to obtain the basic mountain pass solutions themselves, some sort of nondegeneracy condition is generally required and such conditions will be introduced and discussed. The system version of (Per) has been studied in [36]

Two examples

A mountain pass theorem

A third example