Multiplicity and Stability of the Pohozaev Obstruction for Hardy-Schrödinger Equations with Boundary Singularity
Authors: Ghoussoub Nassif , Mazumdar Saikat , Robert Frédéric 
Published: June 2023

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,
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout Enlarged left-brace 1st Row 1st Column minus normal upper Delta u minus gamma StartFraction u Over StartAbsoluteValue x EndAbsoluteValue squared EndFraction minus h left-parenthesis x right-parenthesis u 2nd Column a m p semicolon equals 3rd Column a m p semicolon StartFraction StartAbsoluteValue u EndAbsoluteValue Superscript 2 Super Superscript star Superscript left-parenthesis s right-parenthesis minus 2 Baseline u Over StartAbsoluteValue x EndAbsoluteValue Superscript s Baseline EndFraction 4th Column a m p semicolon in normal upper Omega comma 2nd Row 1st Column u 2nd Column a m p semicolon equals 3rd Column a m p semicolon 0 4th Column a m p semicolon on partial-differential normal upper Omega minus StartSet 0 EndSet comma EndLayout">

{

−
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.