Sublevel sets and global minima of coercive functionals and local minima of their perturbations

Abstract

The aim of the present paper is essentially to prove that if $Φ$ and $Ψ$ are two sequentially weakly lower semicontinuous functionals on a reflexive real Banach space and if $Ψ$ is also continuous and coercive, then then following conclusion holds: if, for some $r > \inf_X Ψ$, the weak closure of the set $Ψ^{-1}(]-\infty, r[)$ has at least $k$ connected components in the weak topology, then, for each $λ> 0$ small enough, the functional $Ψ+ λΦ$ has at least $k$ local minima lying in $Ψ^{-1}(]-\infty, r[)$.