Let
Ω
\Omega
be a smooth bounded domain in
R
n
\mathbb {R}^n
(
n
≥
3
n\geq 3
) such that
0
∈
∂
Ω
0\in \partial \Omega
. We consider issues of non-existence, existence, and multiplicity of variational solutions in
H
1
,
0
2
(
Ω
)
H_{1,0}^2(\Omega )
for the borderline Dirichlet problem,
{
−
Δ
u
−
γ
u
|
x
|
2
−
h
(
x
)
u
a
m
p
;
=
a
m
p
;
|
u
|
2
⋆
(
s
)
−
2
u
|
x
|
s
a
m
p
;
in
Ω
,
u
a
m
p
;
=
a
m
p
;
0
a
m
p
;
on
∂
Ω
∖
{
0
}
,
\begin{equation*} \left \{ \begin {array}{llll} -\Delta u-\gamma \frac {u}{|x|^2}- h(x) u &=& \frac {|u|^{2^\star (s)-2}u}{|x|^s} \ \ &\text {in } \Omega ,\\ \hfill u&=&0 &\text {on }\partial \Omega \setminus \{ 0 \} , \end{array} \right . \end{equation*}
where
0
>
s
>
2
0>s>2
,
2
⋆
(
s
)
≔
2
(
n
−
s
)
n
−
2
{2^\star (s)}≔\frac {2(n-s)}{n-2}
,
γ
∈
R
\gamma \in \mathbb {R}
and
h
∈
C
0
(
Ω
¯
)
h\in C^0(\overline {\Omega })
. We use sharp blow-up analysis on—possibly high energy—solutions of corresponding subcritical problems to establish, for example, that if
γ
>
n
2
4
−
1
\gamma >\frac {n^2}{4}-1
and the principal curvatures of
∂
Ω
\partial \Omega
at
0
0
are non-positive but not all of them vanishing, then Equation (E) has an infinite number of high energy (possibly sign-changing) solutions in
H
1
,
0
2
(
Ω
)
H_{1,0}^2(\Omega )
. This complements results of the first and third authors, who showed in their 2016 article, Hardy-Singular Boundary Mass and Sobolev-Critical Variational Problems, that if
γ
≤
n
2
4
−
1
4
\gamma \leq \frac {n^2}{4}-\frac {1}{4}
and the mean curvature of
∂
Ω
\partial \Omega
at
0
0
is negative, then
(
E
)
(E)
has a positive least energy solution.
On the other hand, the sharp blow-up analysis also allows us to show that if the mean curvature at
0
0
is nonzero and the mass, when defined, is also nonzero, then there is a surprising stability of regimes where there are no variational positive solutions under
C
1
C^1
-perturbations of the potential
h
h
. In particular, and in sharp contrast with the non-singular case (i.e., when
γ
=
s
=
0
\gamma =s=0
), we prove non-existence of such solutions for
(
E
)
(E)
in any dimension, whenever
Ω
\Omega
is star-shaped and
h
h
is close to
0
0
, which include situations not covered by the classical Pohozaev obstruction.
