Localization of critical points in annular conical sets via the method of Nehari manifold

TL;DR

This work addresses the localization of ground-state critical points of a $C^1$ energy functional $E:H\to\mathbb{R}$ within an annular conical region $K_{r,R}$ in a nondegenerate cone $K\subset H$. It combines the Nehari manifold method with a unique scaling $s(u)$ along rays to form a constrained Nehari shell, and then employs Ekeland's variational principle together with an implicit-function argument to obtain a minimizing sequence with $E'(u_n)\to0$; under compactness, this yields a true critical point. The main contributions are the sufficient conditions (the hypotheses (h1)-(h4) and uniform bounds on $E''$) that guarantee existence (and multiplicity) of localized ground states in $K_{r,R}$, and an explicit application to a symmetric Dirichlet problem using a Harnack-type energy estimate to enforce invariance of the Nehari structure. The results extend Nehari-based localization from the whole-domain setting to annular conical domains, providing a practical variational framework for localized nonlinear critical points with potential multiple solutions across disjoint cones.

Abstract

Using the Nehari manifold method, we establish sufficient conditions such that a smooth functional attains a ground state within an annular domain of a closed cone. The localization we obtain immediately allows for multiplicity when applied to disjoint conical sets. To illustrate our results, we consider a two-point boundary value problem and obtain a solution within a shell of a closed cone, defined in terms of a Harnack inequality with respect to the energy norm. The conditions imposed on the nonlinear term naturally extend those from classical examples in the literature which were derived using the method of Nehari manifold on the entire domain.

Theorems & Definitions (15)

• Theorem 1.1: Schechter's Theorem
• Lemma 1.2: Ekeland Principle - weak form
• Theorem 1.3
• Remark 2.1
• Theorem 2.2
• Lemma 2.3
• proof
• proof : Proof of Theorem \ref{['th principala']}
• Theorem 2.4
• proof