Further results related to a minimax problem of Ricceri

We deal with a theoric question raised in connection with the application of a three- critical points theorem, obtained by Ricceri, which has been already applied to obtain multiplicity results for boundary value problems in several recent papers. In the set- tings of the mentioned theorem, the typical assumption is that the following minimax inequality supλ∈I inf x∈X(Φ(x)+λΨ(x)+h(λ)) < inf x∈X supλ∈I(Φ(x)+λΨ(x)+h(λ)) has to be satisfied by some continuous and concave function h: I → R.W henI =(0,+∞(, we have already proved, in a precedent paper, that the problem of finding such function h is equivalent to looking for a linear one. Here, we consider the question for any interval I and prove that the same conclusion holds. It is worth noticing that our main result im- plicitly gives the most general conditions under which the minimax inequality occurs for some linear function. We finally want to stress out that although we employ some ideas similar to the ones developed for the case where I =(0,+∞(, a key technical lemma needs different methods to be proved, since the approach used for that particular case does not work for upper-bounded intervals.