*(English)*Zbl 1187.47057

This paper establishes a new three critical points theorem for the equation

under specific hypotheses. If $X$ is real Banach space, denote by ${\mathcal{W}}_{X}$ the class of functionals ${\Phi}:X\to \mathbb{R}$ possessing the following property: if $\left\{{u}_{n}\right\}$ is a sequence in $X$ converging weakly to $u\in X$ and ${lim\; inf}_{n\to \infty}{\Phi}\left({u}_{n}\right)\le {\Phi}\left(u\right)$, then $\left\{{u}_{n}\right\}$ has a subsequence converging strongly to $u$.

The main result of the paper is as follows.

Theorem 1. Let $X$ be a separable and reflexive real Banach space; $I\subseteq \mathbb{R}$ an interval; ${\Phi}:X\to \mathbb{R}$ a sequentially weakly lower semicontinuous ${C}^{1}$ functional from ${\mathcal{W}}_{X},$ bounded on each bounded subset of $X$ and whose derivative admits a continuous inverse on ${X}^{*}$; $J:X\to \mathbb{R}$ a ${C}^{1}$ functional with compact derivative. Assume that, for each $\lambda \in I$, the functional ${\Phi}-\lambda J$ is coercive and has a strict local, not global minimum, say ${\widehat{x}}_{\lambda}$.

Then, for each compact interval $[a,b]\subseteq I$ for which ${sup}_{\lambda \in [a,b]}({\Phi}\left({\widehat{x}}_{\lambda}\right)-\lambda J\left({\widehat{x}}_{\lambda}\right))<+\infty $, there exists $r>0$ with the following property: for every $\lambda \in [a,b]$ and every ${C}^{1}$ functional ${\Psi}:X\to \mathbb{R}$ with compact derivative, there exists $\delta >0$ such that, for each $\mu \in [0,\delta ]$, the equation

has at least three solutions whose norms are less than $r$.

Some applications of this result are also given.