A further refinement of a three critical points theorem Original Research Article

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 View the MathML sourceΩ⊂Rn (n≥3n≥3) be a bounded domain with smooth boundary and let View the MathML source10ϵ>0 small enough, there exists λϵ>0λϵ>0 such that, for every compact interval View the MathML source[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 View the MathML sourceh:R→R satisfying View the MathML sourcelim sup∣ξ∣→+∞|h(ξ)||ξ|s<+∞ for some View the MathML sources∈]0,n+2n−2[, there exists δ>0δ>0 such that, for each ν∈[0,δ]ν∈[0,δ], the problem View the MathML source{−Δu=ϵ|u|p−1u−λ|u|q−1u+νh(u)in Ωu=0on ∂Ω Turn MathJax on has at least three weak solutions whose norms in View the MathML sourceH01(Ω) are less than ρρ.