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

TL;DR

This paper establishes the existence of positive solutions for a nonlinear elliptic PDE with Sobolev critical exponent under flatness conditions on the coefficient function, providing precise compactness estimates.

Contribution

It introduces a flatness condition on the coefficient function to prove existence results for a Sobolev-critical elliptic equation, extending previous compactness and existence theories.

Findings

Existence of solutions under flatness conditions.

Precise estimates on loss of compactness.

Application of Euler-Hopf type formula.

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.