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Zbl 1214.47079
Ricceri, Biagio
A three critical points theorem revisited.
(English)
[J] Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 70, No. 9, A, 3084-3089 (2009). ISSN 0362-546X

Let $X$ be a reflexive Banach space, $I \subset\Bbb R$ an interval; $\Phi: X \to\Bbb R$ a sequentially weakly lower semicontinuous $C^1$ functional, bounded on each bounded subset of $X$, whose derivative admits a continuous inverse on $X^*$; $J: X \to\Bbb R$ a $C^1$ functional with compact derivative. Assume that $\lim_{\Vert x\Vert \to \infty}(\Phi(x)+\lambda J(x))=+\infty$ for all $\lambda \in I$, and there exists $\rho \in\Bbb R$ 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 exists a subset $A \subset I$, $A \neq \emptyset$, and $r>0$ with the following property: for every $\lambda \in A$ and every $C^1$ functional $\Psi: X \to\Bbb R$ 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 $r$.
[Mikhail Yu. Kokurin (Yoshkar-Ola)]
MSC 2000:
*47J30 Variational methods
58E05 Abstract critical point theory
49J35 Minimax problems (existence)
35J60 Nonlinear elliptic equations

Keywords: minimax theorems; critical points; multiplicity

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