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.