In this paper we obtain an exact result for the extremal value of a class of singular equations with critical exponents. We emphasize that there is no restriction on the shape of the domain. It must be said that the method of sub and supersolutions does not adapt for dealing with estimates of this type, since for general domains (without symmetric property, say) precise information about sub/supersolutions is no longer possible and explicit calculations for extremal values can not be actually carried out. For general domains without symmetric properties it is difficult to derive an exact result for extremal values. Still few general results are known except in [Comm. Partial Differential Equations, 2002 (809-845)] Gazzola and Malchiodi. Research is supported in part by NSFC Grant No.10601063 and 10971238.
Journal of Functional Analysis Volume 260, Number 5/ 2011.3