Electron. J. Diff. Eqns., Vol. 2006(2006), No. 71, pp. 1-17.

Energy quantization for Yamabe's problem in conformal dimension

Fethi Mahmoudi

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 $\mathbb{R}^n$.

Submitted February 20, 2006. Published July 7, 2006.
Math Subject Classifications: 35B33, 46E30, 46L65.
Key Words: Critical exponents; Lorentz spaces; quantization phenomena.

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Fethi Mahmoudi
Scuola Internazionale Superiore Di Studi Avanzati (Sissa)
Via Beirut 2-4, 34014 Trieste, Italy
email: mahmoudi@sissa.it

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