Concentration estimates for Emden-Fowler equations with boundary singularities and critical growth

Abstract

We establish—among other things—existence and multiplicity of solutions for the Dirichlet problem ∑i∂iiu + |u|2*−2u/|x|s = 0 on a smooth bounded domain Ω of ℝn(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 nonsingular framework (i.e., when s = 0), there is no nontrivial solution under global convexity assumption (e.g., when Ω is star-shaped around 0). However, in contrast to the nonsatisfactory situation of the nonsingular 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 nonpositive 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 Ghoussoub and Kang in 2004 and completing our result in 2005 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 nonsingular case.