A three critical points theorem revisited

In this paper the following result is proved: Let $X$ be a reflexive real Banach space; $I\subseteq {\bf R}$ an interval; $\Phi:X\to {\bf R}$ a sequentially weakly lower semicontinuous $C1$ functional, bounded on each bounded subset of $X$, whose derivative admits a continuous inverse on $X^*$; $J:X\to {\bf R}$ a $C1$ functional with compact derivative. Assume that $$\lim_{\|x\|\to +\infty}(\Phi(x)+\lambda J(x))=+\infty $$ for all $\lambda\in I$, and that there exists $\rho\in {\bf R}$ such that $$\sup_{\lambda\in I}\inf_{x\in X}(\Phi(x)+ \lambda(J(x)+\rho))<\inf_{x\in X} \sup_{\lambda\in I}(\Phi(x)+\lambda(J(x)+\rho))\ .$$ Then, there exist a non-empty open set $A\subseteq I$ and a positive real number $r$ with the following property: for every $\lambda\in A$ and every $C1$ functional $\Psi:X\to {\bf R}$ with compact derivative, there exists $\delta>0$ such that, for each $\mu\in [0,\delta]$, the equation $$\Phi'(x)+\lambda J'(x)+\mu\Psi'(x)=0$$ has at least three solutions in $X$ whose norms are less than $r$