Critical point localization and multiplicity results in Banach spaces via Nehari manifold technique

TL;DR

This paper introduces a novel approach combining the Nehari manifold technique with Birkhoff-Kellogg's theorem to establish critical point existence and multiplicity in Banach spaces, simplifying traditional variational methods.

Contribution

It presents a new method for locating critical points and proving multiplicity in Banach spaces, using a combination of Nehari manifold and invariant-direction theorems, offering a simpler alternative to existing approaches.

Findings

Established existence of critical points in annular subsets of cones.

Proved multiplicity results for solutions of boundary value problems.

Demonstrated the method on p-Laplacian equations.

Abstract

In the paper, results on the existence of critical points in annular subsets of a cone are obtained with the additional goal of obtaining multiplicity results. Compared to other approaches in the literature based on the use of Krasnoselskii's compression-extension theorem or topological index methods, our approach uses the Nehari manifold technique in a surprising combination with the cone version of Birkhoff-Kellogg's invariant-direction theorem. This yields a simpler alternative to traditional methods involving deformation arguments or Ekeland variational principle. The new method is illustrated on a boundary value problem for p-Laplacian equations, and we believe that it will be useful for proving the existence, localization, and multiplicity of solutions for other classes of problems with variational structure.