We establish -among other things-existence and multiplicity of solutions for the Dirichlet problem P i ∂ ii u+ |u| 2 −2 u |x| s = 0 on smooth bounded domains Ω of R n (n ≥ 3) involving the critical Hardy-Sobolev exponent 2 = 2(n−s) n−2 where 0 < s < 2, and in the case where zero (the point of singularity) is on the boundary ∂Ω. Just as in the Yamabe-type non-singular framework (i.e., when s = 0), there is no nontrivial solution under global convexity assumption (e.g., when Ω is star-shaped

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... d 0). However, in contrast to the nonsatisfactory situation of the non-singular case, we show the existence of an infinite number of solutions under an assumption of local strict concavity of ∂Ω at 0 in at least one direction. More precisely, we need the principal curvatures of ∂Ω at 0 to be non-positive but not all vanishing. We also show that the best constant in the Hardy-Sobolev inequality is attained as long as the mean curvature of ∂Ω at 0 is negative, extending the results of [21] and completing our result of [22] to include dimension 3. The key ingredients in our proof are refined concentration estimates which yield compactness for certain Palais-Smale sequences which do not hold in the non-singular case. Date: March, 13th 2005. Both authors gratefully acknowledge the support of the Natural Sciences and Engineering Research Council of Canada and the hospitality of the University of British Columbia where this work was initiated.