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

Abstract

In this paper, we consider the following nonlinear elliptic equation with Dirichlet 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 domain in $\mathbb{R}^n,$ $n\geqslant 4,$ and $K$ is a $\mathcal{C}^1$-positive function in $\bar{\Omega}$. Under the assumption that the order of flatness at each critical point of $K$ is $\beta \in ]\,n-2,\,n[,$ we give precise estimates on the looses of the compactness, and we prove an existence result through an Euler-Hopf type formula.