Concentration and multiplicity of solutions for a fourth-order equation with critical nonlinearity

In this paper, we consider the problem (Pε)(Pε) : Δ2u=un+4/n-4+εu,u>0Δ2u=un+4/n-4+εu,u>0 in Ω,u=Δu=0Ω,u=Δu=0 on ∂Ω∂Ω, where ΩΩ is a bounded and smooth domain in Rn,n>8Rn,n>8 and ε>0ε>0. We analyze the asymptotic behavior of solutions of (Pε)(Pε) which are minimizing for the Sobolev inequality as ε→0ε→0 and we prove existence of solutions to (Pε)(Pε) which blow up and concentrate around a critical point of the Robin's function. Finally, we show that for εε small, (Pε)(Pε) has at least as many solutions as the Ljusternik–Schnirelman category of ΩΩ.