We investigate the Hardy–Schrödinger operator
Lγ=−Δ−γ/∣∣x∣∣2 on smooth domains
Ω⊂Rn whose boundaries
contain the singularity
0.
We prove a Hopf-type result and optimal regularity for variational solutions of
corresponding linear and nonlinear Dirichlet boundary value problems, including the
equation
Lγu=u2⋆(s)−1/∣∣x∣∣s,
where
γ<14n2,
s∈[0,2) and
2⋆(s):=2(n−s)/(n−2) is
the critical Hardy–Sobolev exponent. We also give a complete description
of the profile of all positive solutions — variational or not — of the
corresponding linear equation on the punctured domain. The value
γ=14(n2−1)
turns out to be a critical threshold for the operator
Lγ. When
14(n2−1)<γ<14n2, a notion of
Hardysingular boundary mass mγ(Ω)
associated to the operator
Lγ
can be assigned to any conformally bounded
domain Ω
such that
0∈∂Ω.
As a byproduct, we give a complete answer to problems of existence of extremals for
Hardy–Sobolev inequalities, and consequently for those of Caffarelli, Kohn and
Nirenberg. These results extend previous contributions by the authors in the case
γ=0, and by Chern and
Lin for the case
γ<14(n−2)2.
More specifically, we show that extremals exist when
0≤γ≤14(n2−1) if the mean
curvature of
∂Ω at
0 is negative. On
the other hand, if
14(n2−1)<γ<14n2,
extremals then exist whenever the Hardy singular boundary mass
mγ(Ω) of the
domain is positive.
Keywords
Hardy–Schrödinger operator,
Hardy-singular boundary mass, Hardy–Sobolev
inequalities, mean curvature