On the Hardy-Schrödinger operator with a singularity on the boundary

Abstract

We consider borderline elliptic partial differential equations involving the Hardy-Schrödinger $L_{\gamma }:=-\Delta -\gamma \frac{1}{\left|x\right|{}^2}$ operator on a domain $\Omega \subset R^n$, when the singularity zero is on the boundary of the domain. This operator arises naturally when dealing with the Caffarelli-Kohn-Nirenberg inequalities and their associated Euler-Lagrange equations.

Now, it is well known that the operator $L_{\gamma }$ is non-negative when $0$ is in the interior of a domain as long as $\gamma \le \frac{{\left(n-2\right)}^2}{4}$. The situation is much more interesting when $0\in \partial \Omega$. For one, the operator is then non-negative for all $\gamma \le \frac{n^2}{4}$, at least for convex domains. The problem of whether the Dirichlet boundary problem

has positive solutions is closely related to whether the best constants in the Caffarelli-Kohn-Nirenberg inequalities are attained. Here $2^{*}\left(s\right)=\frac{2\left(n-s\right)}{n-2}$ and $s\in \left[0,2\right)$. Brezis-Nirenberg type methods were used by C.S. Lin et al. to show that this is indeed the case when $\gamma <\frac{{\left(n-2\right)}^2}{4}$ under the condition that the mean curvature of the domain at $0$ is negative. Their results extend previous work by Ghoussoub-Robert who dealt with the case $\gamma =0$.

The case when $\frac{{\left(n-2\right)}^2}{4}\le \gamma <\frac{n^2}{4}$ turned out to be much more delicate. A detailed analysis of the linear Hardy-Schrodinger operator $L\gamma$ performed recently by Ghoussoub-Robert surprisingly show that $\gamma =\frac{n^2-1}{4}$ is another critical threshold for the operator. While the C. S. Lin et al. results extend to the situation where $\gamma <\frac{n^2-1}{4}$, the interval $\gamma \in \left[\frac{n^2-1}{4},\frac{n^2}{4}\right)$ requires the introduction of a notion of ”mass” in the spirit of Shoen-Yau for the Hardy-Schrödinger operator. The existence of solutions then depend on the sign of such a mass. Examples of the ”zoology” of the sign of the mass will be given too.

(joint work with Nassif Ghoussoub).