Critical point theorems (Q1607870)

One of the main results of the paper states: Theorem. Let \(H\) be a Hilbert space such that \(H=V\oplus W\) where \(V\) and \(W\) are closed and orthogonal subspaces of \(H\). Let \(\Phi: H\to \mathbb{R}\) be a functional such that (i) \(\Phi\) is of class \(\mathcal C^2\). (ii) There exists a continuous nonincreasing function \(\gamma: [0,+\infty)\to (0,+\infty)\) such that \[ \langle D^2\Phi(v+w)g,g\rangle \leq -\gamma(\|v\|)\|g\|^2 \] for all \(v\in V, w\in W\) and \(g\in V\). (iii) \(\Phi\) is coercive on \(W\). (iv) For all \(w\in W\), \(\Phi(v+w) \to -\infty\) when \(\|v\|\to +\infty\), \(v\in V\). (v) \(\Phi\) is weakly lower semicontinuous on \(W+v\). Then \(\Phi\) admits at least a critical point \(u\in H\). Moreover, this critical point of \(\Phi\) is characterized by the equality \[ \Phi(u) = \min_{w\in W}\max_{v\in V} \Phi(v+w). \] This theorem is a generalization of a result of \textit{A. C. Lazer, E. M. Landesman} and \textit{D. R. Meyers} [J. Math. Anal. Appl. 52, 594-614 (1975; Zbl 0354.35004)].