  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
$\Psi^{-1}(]-\infty, r[)$.
