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

In this paper we consider the fourth-order problem { Δ 2 u = μ | u | s − 1 u + | u | 2 ⁎ − 2 u in Ω , u , − Δ u > 0 in Ω , u , Δ u = 0 on ∂ Ω , where Ω is a smooth bounded domain in R N , N ≥ 5 and 2 ⁎ = 2 N / ( 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 cat Ω ( Ω ) , in case the parameter μ > 0 is sufficiently small.