Instability for standing waves of nonlinear Klein-Gordon equations via mountain-pass arguments

TL;DR

This paper employs mountain-pass arguments to analyze the orbital instability of standing waves in nonlinear Klein-Gordon equations, simplifying proofs and extending classical instability results to new settings.

Contribution

It introduces mountain-pass techniques to study orbital instability, providing new proofs and extending classical results to two-dimensional cases.

Findings

Ground states are minimizers of the action functional under various constraints.

Extended instability results to the two-dimensional case for general nonlinearities.

Simplified proofs of orbital instability using mountain-pass arguments.

Abstract

We introduce mountain-pass type arguments in the context of orbital instability for Klein-Gordon equations. Our aim is to illustrate on two examples how these arguments can be useful to simplify proofs and derive new results of orbital stability/instability. For a power-type nonlinearity, we prove that the ground states of the associated stationary equation are minimizers of the functional "action" on a wide variety of constraints. For a general nonlinearity, we extend to the dimension N=2 the classical instability result for stationary solutions of nonlinear Klein-Gordon equations proved in 1985 by Shatah in dimension N>2.