A further refinement of a three critical points theorem

In this paper, we complete the refinement process, made by Ricceri (2009) [4], of a result established by Ricceri (2000) [1], which is one of the most applied abstract multiplicity theorems in the past decade. A sample of application of our new result is as follows.Let Ω⊂Rn (n≥3n≥3) be a bounded domain with smooth boundary and let 10ϵ>0 small enough, there exists λϵ>0λϵ>0 such that, for every compact interval [a,b]⊂]0,λϵ[, there exists ρ>0ρ>0 with the following property: for every λ∈[a,b]λ∈[a,b] and every continuous function h:R→R satisfying lim sup∣ξ∣→+∞|h(ξ)||ξ|s<+∞ for some s∈]0,n+2n−2[, there exists δ>0δ>0 such that, for each ν∈[0,δ]ν∈[0,δ], the problem {−Δu=ϵ|u|p−1u−λ|u|q−1u+νh(u)in Ωu=0on ∂Ω has at least three weak solutions whose norms in H01(Ω) are less than ρρ.