The method of the energy function and applications

TL;DR

This paper introduces a new method using an energy function to find critical points of differentiable functionals in Banach spaces, enabling solutions to variational problems without the Ambrosetti-Rabinowitz condition.

Contribution

The paper develops a novel approach linking energy functions to critical points of functionals, extending variational methods to broader classes of problems.

Findings

Established a new method for critical point analysis in Banach spaces.

Applied the method to solve variational elliptic problems.

Provided a mountain pass theorem variant without the Ambrosetti-Rabinowitz condition.

Abstract

In this work, we establish a new method to find critical points of differentiable functionals defined in Banach spaces which belong to a suitable class ($\mathcal{J}$) of functionals. Once given a functional $J$ in the class ($\mathcal{J}$), the central idea of the referred method consists in defining a real function $\zeta$ of a real variable, called {\it energy function}, which is naturally associated to $J$ in the sense that the existence of real critical points for $\zeta$ guarantees the existence of critical points for the functional $J$. As a consequence, we are able to solve some variational elliptic problems, whose associated energy functional belongs to ($\mathcal{J}$) and provide a version of the mountain pass theorem for functionals in the class ($\mathcal{J}$) that allows us to obtain mountain pass solutions without the so-called Ambrosetti-Rabinowitz condition.