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

TL;DR

The paper develops a novel method to localize and obtain multiple critical points of variational functionals in annular conical regions of Banach spaces by blending Nehari-manifold techniques with a cone version of the Birkhoff-Kellogg invariant-direction theorem, avoiding traditional deformation arguments and Ekeland's principle. It extends prior Hilbert-space results to Banach spaces and requires only C^1 regularity. The approach yields existence and multiplicity results for boundary-value problems with variational structure, demonstrated on p-Laplacian equations and facilitated by a Harnack-type inequality and duality-mapping framework. This provides a streamlined, robust tool for proving localized, multiple solutions in cones and potentially extends to broader nonlinear operators.

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.

Theorems & Definitions (16)

• Proposition 2.1
• Lemma 2.2: lpv
• Theorem 2.3: Krasnoselskii and Ladyzenskii
• Lemma 3.1
• proof
• Theorem 3.2
• proof
• Theorem 3.3: Multiplicity
• proof
• Theorem 4.1