nergy quantization for Yamabe's problem in conformal dimension

Abstract

Riviere [11] proved an energy quantization for Yang-Mills fields defined on $n$-dimensional Riemannian manifolds, when $n$ is larger than the critical dimension 4. More precisely, he proved that the defect measure of a weakly converging sequence of Yang-Mills fields is quantized, provided the $W^{2,1}$ norm of their curvature is uniformly bounded. In the present paper, we prove a similar quantization phenomenon for the nonlinear elliptic equation $$ - Delta{u}= u |u|^{4/(n-2)}, $$ in a subset $Omega$ of $mathbb{R}^n$.

Keywords

• Critical exponents
• Lorentz spaces
• quantization phenomena.