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In this paper, we consider the following nonlinear elliptic equation withDirichlet boundary condition: $-\Delta u=K(x)u^{\frac{n+2}{n-2}},\, u>0$ in$\Omega,\, u=0$ on $\partial\Omega$, where $\Omega$ is a smooth bounded domainin $\mathbb{R}^n,$ $n\geqslant 4,$ and $K$ is a $\mathcal{C}^1$-positivefunction in $\bar{\Omega}$. Under the assumption that the order of flatness ateach critical point of $K$ is $\beta \in ]\,n-2,\,n[,$ we give preciseestimates on the looses of the compactness, and we prove an existence resultthrough an Euler-Hopf type formula.

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