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

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