Singular limits for the bi-Laplacian operator with exponential nonlinearity in R4R4

Let Ω   be a bounded smooth domain in R4R4 such that for some integer d⩾1d⩾1 its d-th singular cohomology group with coefficients in some field is not zero, then problem{Δ2u−ρ4k(x)eu=0in Ω,u=Δu=0on ∂Ω, has a solution blowing-up, as ρ→0ρ→0, at m points of Ω, for any given number m.