A generalized mountain-pass theorem: the existence of multiple critical points

Abstract

We analyse the existence of multiple critical points for an even functional $$\begin{aligned} J:H\rightarrow {\mathbb {R}}\end{aligned}$$ in the following context: the Hilbert space H can be split into an orthogonal sum \(H = Y \oplus Z\) in such a way that $$\begin{aligned} \inf \{ J(u) : u \in Z \text { and } \Vert u\Vert = \rho \} \ge \alpha > J(0) \end{aligned}$$ and that here exits a point \(b \in H\) with \(\Vert b\Vert > \rho \) and with \(J(b) \le J(0)\). We develop a new variational characterization of multiple critical levels without an assumption on the dimension of Y. Our characterization is simple and natural: we can for example avoid the notion of pseudo-index and the definition of the activated levels does not substantially differs from the one used for the lowest critical level, giving us in this way a unified view of critical levels. We apply our results to a semi-linear Schrodinger equation of the form $$\begin{aligned} \left\{ \begin{array}{ll} &{} -\Delta u + V (x) u - q (x) |u|^ \sigma u = \lambda u, \qquad x\in {\mathbb {R}}^ N\\ &{} u \in {H^1({\mathbb {R}}^N)}\setminus \{ 0 \} \end{array} \right. \end{aligned}$$ where \(\lambda \) is inside a spectral gap bounded on both sides by parts of the essential spectrum.