Zero energy critical points of functionals depending on a parameter

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.