We consider borderline elliptic partial differential equations involving the Hardy-Schrödinger Lγ:=-Δ-γ1|x|2 operator on a domain Ω⊂Rn, 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 Lγ is non-negative when 0 is in the interior of a domain as long as γ≤(n-2)24. The situation is much more interesting when 0∈∂Ω. For one, the operator is then non-negative for all γ≤n24, at least for convex domains. The problem of whether the Dirichlet boundary problem
| -Δu-γ|x|2u=u2*(s)-1|x|s on Ω | (1) |
has positive solutions is closely related to whether the best constants in the Caffarelli-Kohn-Nirenberg inequalities are attained. Here 2*(s)=2(n-s)n-2 and s∈[0,2). Brezis-Nirenberg type methods were used by C.S. Lin et al. to show that this is indeed the case when γ<(n-2)24 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 γ=0.
The case when (n-2)24≤γ<n24 turned out to be much more delicate. A detailed analysis of the linear Hardy-Schrodinger operator Lγ performed recently by Ghoussoub-Robert surprisingly show that γ=n2-14 is another critical threshold for the operator. While the C. S. Lin et al. results extend to the situation where γ<n2-14, the interval γ∈[n2-14,n24) 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).