On a fourth-order elliptic equation involving the critical Sobolev exponent: The effect of the graph topology

In this paper, we consider the following nonlinear equation Δ2u=Kun+4n−4+εu, with u>0u>0 in ΩΩ and u=Δu=0u=Δu=0 on ∂Ω∂Ω, where ΩΩ is a smooth bounded domain in RnRn, n>8n>8, KK is a smooth positive function in ΩΩ and εε is a small positive parameter. Using the effect of the graph topology of the function KK, we prove existence, multiplicity and concentration of solutions. We prove also a nonexistence result.