PDE Seminar Abstracts

Frédéric Robert

Université de Lorraine, France

Tuesday 20 January 2015, 2-3pm, Carslaw Room 829 (AGR)

Université de Lorraine, France

Tuesday 20 January 2015, 2-3pm, Carslaw Room 829 (AGR)

We consider borderline elliptic partial differential equations involving the Hardy-Schrödinger ${L}_{\gamma}:=-\Delta -\gamma \frac{1}{\left|x\right|\phantom{\rule{-0.2pt}{0ex}}{\text{}}^{2}}$ 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).

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