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

TLDR: In this article, it was shown that if π and π are two sequentially weakly lower semicontinuous functionals on a reflexive real Banach space, then the weak closure of the set π^{-1}(]-\infty, r[)$ has at least $k$ connected components in the weak topology.

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