Local minimizers for a class of functionals over the Nehari set

TL;DR

This paper investigates the structure of the Nehari set for certain functionals, focusing on the extremal parameter where the set becomes a natural constraint, and establishes the existence of local minimizers even when the functional is unbounded below.

Contribution

It extends and unifies previous results by analyzing local minimizers on the Nehari set for a broad class of functionals with applications to various differential problems.

Findings

Identifies the extremal parameter $mbda^*$ where the Nehari set is a natural constraint.

Proves existence of local minimizers when the energy functional is unbounded below.

Unifies previous results for indefinite, $(p,q)$-Laplacian, and Kirchhoff problems.

Abstract

We analyze the topological structure of the Nehari set for a class of functionals depending on a real parameter $\lambda$, and having two degrees of homogeneity. A special attention is paid to the extremal parameter $\lambda^*$, which is the threshold value for the Nehari set to be given by a {\it natural constraint}. The main difficulty arises when $\lambda>\lambda^*$, as the energy functional may be unbounded from below over the Nehari set. In such situation we prove the existence of local minimizers of the functional constrained to this set. We unify and extend previous existence and multiplicity results for critical points of indefinite, $(p,q)$-Laplacian, and Kirchhoff type problems.