Positive solutions to a fourth-order elliptic problem by the Lusternik–Schnirelmann category

Abstract

In this paper we consider the fourth-order problem{Δ2u=μ|u|s−1u+|u|2⁎−2uinΩ,u,−Δu>0in Ω,u,Δu=0on ∂Ω, where Ω is a smooth bounded domain in RN, N≥5 and 2⁎=2N/(N−4). We assume 2≤s+1<2⁎ in case N≥8 and 2⁎−2<s+1<2⁎ for the critical dimensions N=5,6,7. Then we prove that if Ω has a rich topology, described by its Lusternik–Schnirelmann category, then the problem has multiple solutions, at least as many as catΩ(Ω), in case the parameter μ>0 is sufficiently small.