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 $\sum_i\partial_{ii}u+\frac{|u|^{\crit-2}u}{|x|^s}=0$ on smooth bounded domains $\Omega$ of $ \rn$ ($n\geq 3$) involving the critical Hardy-Sobolev exponent $\crit =\frac{2(n-s)}{n-2}$ where $0<s<2$, and in the case where zero (the point of singularity) is on the boundary $\partial \Omega$. 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 $\Omega$ is star-shaped around 0). However, in contrast to the non-satisfactory situation of the non-singular case, we show the existence of an infinite number of solutions under an assumption of local strict concavity of $\partial \Omega$ at 0 in at least one direction. More precisely, we need the principal curvatures of $\partial \Omega$ 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 $\partial \Omega$ 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.