Zero energy critical points of functionals depending on a parameter

Abstract

We investigate zero energy critical points for a class of functionals Φµ defined on a uniformly convex Banach space, and depending on a real parameter µ. More precisely, we show the existence of a sequence (µn) such that Φµn has a pair of critical points ±un satisfying Φµn (±un) = 0, for every n. In addition, we provide some properties of µn and un. This result, which is proved by combining the nonlinear generalized Rayleigh quotient method [10] with the Ljusternik-Schnirelman theory, is then applied to several classes of elliptic pdes.