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A further three critical points theorem. (English) Zbl 1187.47057

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

${{\Phi }}^{\text{'}}\left(x\right)=\lambda {J}^{\text{'}}\left(x\right)+\mu {{\Psi }}^{\text{'}}\left(x\right)$

under specific hypotheses. If $X$ is real Banach space, denote by ${𝒲}_{X}$ the class of functionals ${\Phi }:X\to ℝ$ 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; $I\subseteq ℝ$ an interval; ${\Phi }:X\to ℝ$ a sequentially weakly lower semicontinuous ${C}^{1}$ functional from ${𝒲}_{X},$ bounded on each bounded subset of $X$ and whose derivative admits a continuous inverse on ${X}^{*}$; $J:X\to ℝ$ 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 ${\stackrel{^}{x}}_{\lambda }$.

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

${{\Phi }}^{\text{'}}\left(x\right)=\lambda {J}^{\text{'}}\left(x\right)+\mu {{\Psi }}^{\text{'}}\left(x\right)$

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

Some applications of this result are also given.

##### MSC:
 47J30 Variational methods (nonlinear operator equations) 58E05 Abstract critical point theory 49J35 Minimax problems (existence) 35J60 Nonlinear elliptic equations