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Zbl 1187.47057
Ricceri, Biagio
A further three critical points theorem.
(English)
[J] Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 71, No. 9, A, 4151-4157 (2009). ISSN 0362-546X

This paper establishes a new three critical points theorem for the equation $$\Phi'(x)=\lambda J'(x)+ \mu\Psi'(x)$$ under specific hypotheses. If $X$ is real Banach space, denote by $\mathcal{W}_{X}$ the class of functionals $\Phi:X\rightarrow\mathbb{R}$ possessing the following property: if $\{u_n\}$ is a sequence in $X$ converging weakly to $u\in X$ and $\liminf_{n\to\infty} \Phi(u_n)\le \Phi(u)$, then $\{u_n\}$ 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; $I\subseteq\Bbb R$ an interval; $\Phi:X\to\Bbb R$ a sequentially weakly lower semicontinuous $C^1$ functional from ${\cal W}_X,$ bounded on each bounded subset of $X$ and whose derivative admits a continuous inverse on $X^*$; $J:X\to\Bbb R$ 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 $\sup_{\lambda\in[a,b]} (\Phi(\widehat{x}_\lambda)- \lambda J(\widehat{x}_\lambda))< +\infty$, there exists $r>0$ with the following property: for every $\lambda\in[a,b]$ and every $C^1$ functional $\Psi :X\to\Bbb R$ with compact derivative, there exists $\delta>0$ such that, for each $\mu\in[0,\delta]$, the equation $$\Phi'(x)= \lambda J'(x)+ \mu\Psi'(x)$$ has at least three solutions whose norms are less than $r$. Some applications of this result are also given.
[N. C. Apreutesei (Iaşi)]
MSC 2000:
*47J30 Variational methods
58E05 Abstract critical point theory
49J35 Minimax problems (existence)
35J60 Nonlinear elliptic equations

Keywords: strict local minimum; critical point; multiplicity; nonlinear elliptic equation

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