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

摘要

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