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

12 pages

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