3. Proof of theorem 1.1

The key argument in the proof of our main result is a three-critical-point theorem due to Ricceri [30]. This result is widely applied to solve various elliptic problems; we refer the reader to [4–6, 20, 31]. Ricceri’s result goes back to an elementary property established by Pucci and Serrin (see [30, theorem 3]) which asserts that if a functional of class C1 defined on a real Banach space has two local minima, then it has a third critical point. This is an auxiliary result related to a problem of Rabinowitz [28], who raised the question whether critical points of mountainpass type must necessarily be saddle points. To the best of our knowledge, the first three-critical-point property was found by Krasnoselskii [18]; he showed that if f is a coercive C1 functional defined on a finite-dimensional space having a nondegenerate critical point x0 (that is, the topological index ind f (x0)(0) is non-zero) which is not a global minimum, then f admits a third critical point. This result was extended to infinite-dimensional Banach spaces by Amann [3].