The Effect of the Shape of the Domain on the Existence of Solutions of an Equation Involving the Critical Sobolev Exponent

We study the critical Sobolev exponent equation. By imposing some conditions on the shape of the domain we are able to obtain existence and multiplicity of positive solutions. In particular, multiplicity results for contractible domains are presented, based on an approach that allows us to generalize previous work by Ding, who had proved an existence result for a contractible domain