Abstract
Let E be a real separable and reflexive Banach space, X⊆E weakly closed and unbounded, Φ and Ψ two non-constant weakly sequentially lower sernicontinuous functionals defined on X, such that Φ+λΨ is coercive for each λ≥0. In this setting, if
supλ≥0infx∈X(Φ(x)+λ(Ψ(x)+ρ))=infx∈Xsupλ≥0(Φ(x)+λ(Ψ(x)+ρ))
for every ρ∈R, then, one has
supλ≥0infx∈X(Φ(x)+λΨ(x)+h(λ))=infx∈Xsupλ≥0(Φ(x)+λΨ(x)h(λ)),
for every concave function h:[0,+∞[→R.