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 ${\mathbb{R}}^n,$ $n⩾4,$ 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.

29 pages