A further three critical points theorem

If $X$ is a real Banach space, we denote by ${\cal W}_X$ the class of all functionals $\Phi:X\to {\bf R}$ possessing the following property: if $\{u_n\}$ is a sequence in $X$ converging weakly to $u\in X$ and $\liminf_{n\to \infty}\Phi(u_n)\leq \Phi(u)$, then $\{u_n\}$ has a subsequence converging strongly to $u$. In this paper, we prove the following result: Let $X$ be a separable and reflexive real Banach space; $I\subseteq {\bf R}$ an interval; $\Phi:X\to {\bf R}$ a sequentially weakly lower semicontinuous $C^1$ functional, belonging to ${\cal W}_X$, bounded on each bounded subset of $X$ and whose derivative admits a continuous inverse on $X^*$; $J:X\to {\bf R}$ 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 $\hat x_{\lambda}$.\par Then, for each compact interval $[a,b]\subseteq I$ for which $\sup_{\lambda\in [a,b]}(\Phi(\hat x_{\lambda}) -\lambda J(\hat x_{\lambda}))<+\infty$, there exists $r>0$ with the following property: for every $\lambda\in [a,b]$ and every $C^1$ functional $\Psi:X\to {\bf R}$ with compact derivative, there exists $\delta>0$ such that, for each $\mu\in [0,\delta]$, the equation $$\Phi'(x)=\lambda J'(x)+\mu\Psi'(x)$$ has at least three solutions whose norms are less than $r$.