Zero energy critical points of functionals depending on a parameter

TL;DR

This paper studies the existence of zero energy critical points for parameter-dependent functionals in Banach spaces, using variational methods, and applies the results to elliptic PDEs.

Contribution

It introduces a novel approach combining fibering maps and Ljusternik-Schnirelman theory to find critical points with zero energy for a class of functionals.

Findings

Existence of sequences of parameters with critical points at zero energy.

Properties of the critical points and parameters are characterized.

Application to various classes of elliptic PDEs demonstrates the method's versatility.

Abstract

We investigate zero energy critical points for a class of functionals $\Phi_\mu$ defined on a uniformly convex Banach space, and depending on a real parameter $\mu$. More precisely, we show the existence of a sequence $(\mu_n)$ such that $\Phi_{\mu_n}$ has a pair of critical points $\pm u_n$ satisfying $\Phi_{\mu_n}(\pm u_n)=0$, for every $n$. In addition, we provide some properties of $\mu_n$ and $u_n$. This result, which is proved via a fibering map approach (based on the {\it nonlinear generalized Rayleigh quotient} method \cite{I1}) combined with the Ljusternik-Schnirelman theory, is then applied to several classes of elliptic pdes.