Existence result under flatness condition for a nonlinear elliptic equation with Sobolev exponent

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In this paper, we consider the following nonlinear elliptic equation with Dirichlet boundary condition: -Δu=K(x)uⁿ⁺²/ⁿ⁻², u>0 in Ω, u=0 on ∂Ω, where Ω is a smooth bounded domain in Rⁿ, n 4, and K is a C¹-positive function in Ω. Under the assumption that the order of flatness at each critical point of K is β∈]n-2,n[, we give precise estimates on the looses of the compactness, and we prove an existence result through an Euler-Hopf type formula.