Second order information of Palais-Smale sequences in the mountain pass theorem (Q1197354)

The authors consider a \(C^2\) functional \(\varphi\) defined on a Hilbert space \(H\) such that the conditions of the Mountain Pass theorem are satisfied. They construct a Palais-Smale sequence \((x_n)\) such that: If \(E\subset H\) is a subspace and \(\langle\varphi''(x_n)u,u\rangle < - \frac{1}{n}\| u\|^ 2\) for every \(u\in E\), then \(\dim E\leq 1\). If the Hessian of \(\varphi\) is non-degenerate, then any cluster point for \((x_n)\) is a critical point of Morse index at most one.