Convergence of a mountain pass type algorithm for strongly indefinite problems and systems

TL;DR

This paper analyzes a mountain pass algorithm for strongly indefinite problems, proving convergence to critical points under certain conditions, with applications to Schrödinger equations and systems.

Contribution

It introduces a convergence analysis for a mountain pass algorithm tailored to strongly indefinite problems, including step size strategies and localization assumptions.

Findings

Convergence up to a subsequence to a critical point is established.

Whole sequence convergence is achieved under localization assumptions.

Applications to indefinite Schrödinger equations and systems demonstrate practical relevance.

Abstract

For a functional $\E$ and a peak selection that picks up a global maximum of $\E$ on varying cones, we study the convergence up to a subsequence to a critical point of the sequence generated by a mountain pass type algorithm. Moreover, by carefully choosing stepsizes, we establish the convergence of the whole sequence under a "localization" assumption on the critical point. We illustrate our results with two problems: an indefinite Schr\"odinger equation and a superlinear Schr\"odinger system.