Concentration Estimates for Emden-Fowler Equations with Boundary Singularities and Critical Growth

TL;DR

This paper proves the existence and multiplicity of solutions for a class of Emden-Fowler equations with boundary singularities and critical growth, highlighting the role of boundary curvature and concentration estimates.

Contribution

It introduces new existence results for solutions with boundary singularities under local boundary curvature conditions, extending previous non-singular case results.

Findings

Existence of infinitely many solutions under local strict concavity of boundary.

Best constant in Hardy-Sobolev inequality is attained when boundary mean curvature is negative.

Refined concentration estimates enable compactness in critical boundary singularity problems.

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.