Further results related to a minimax problem of Ricceri
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We deal with a theoric question raised in connection with the application of a threecritical points theorem, obtained by Ricceri, which has been already applied to obtain multiplicity results for boundary value problems in several recent papers. In the settings 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. When I = [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 implicitly 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. 1. Introduction Here and throughout the sequel, E is a real separable and reflexive Banach space, X is a weakly closed unbounded subset of E, I ⊆ R an interval and Φ, Ψ are two (nonconstant) sequentially weakly lower semicontinuous functionals on X such that lim
x∈X, x→+∞
Φ(x) + λΨ(x) = +∞
(1.1)
for all λ ∈ I. In these settings, Ricceri showed that if there exists a continuous concave function h : I → R such that
sup inf Φ(x) + λΨ(x) + h(λ) < inf sup Φ(x) + λΨ(x) + h(λ) , λ∈I x∈X
(1.2)
x∈X λ∈I
then there is an open interval J ⊆ I such that, for each λ ∈ J, the functional Φ + λΨ has Copyright © 2005 Hindawi Publishing Corporation Journal of Inequalities and Applications 2005:5 (2005) 523–533 DOI: 10.1155/JIA.2005.523
524
Further results related to a minimax problem of Ricceri
a local nonabsolute minimum in the relative weak topology of X [11, 12, 13]. Under further assumptions, this fact leads to a three critical points theorem (see [13, Theorem 1] improving [12, Theorem 3.1]) which has been widely applied to get multiplicity results for nonlinear boundary value problems [1, 2, 3, 5, 6, 8, 9, 10, 11, 12, 13]. A natural way to get (1.2) is by a linear function. In view of applications of [12, Theorem 3.1], Ricceri gave useful conditions (see [12, Proposition 3.1]) under which (1.2), with I = [0,+∞[, is satisfied by some linear function (see also [4]). In the same paper, Remark 5.2, Ricceri asked if (1.2) could be satisfied by a suitable continuous concave function also when this does not happen for linear ones. A complete and negative answer was given in [7, Theorem 1] but when the interval I is [0,+∞[. It is still an open and nontrivial problem if the same conclusion would hold for any interval I ⊆ R. In this paper, an answer to this question is given. Before our main result is stated, some notations are needed to be fixed. Let α ∈ R a
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