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research
Sign-Changing Solutions for Critical Equations with Hardy Potential
Authors
Pierpaolo Esposito
Nassif Ghoussoub
Angela Pistoia
Giusi Vaira
Publication date
1 January 2017
Publisher
View
on
arXiv
Abstract
We consider the following perturbed critical Dirichlet problem involving the Hardy-Schr\"odinger operator on a smooth bounded domain
Ω
⊂
R
N
\Omega \subset \mathbb{R}^N
Ω
⊂
R
N
,
N
≥
3
N\geq 3
N
≥
3
, with
0
∈
Ω
0 \in \Omega
0
∈
Ω
:
{
−
Δ
u
−
γ
u
∣
x
∣
2
−
ϵ
u
=
∣
u
∣
4
N
−
2
u
inÂ
Ω
u
=
0
onÂ
∂
Ω
,
\left\{ \begin{array}{ll}-\Delta u-\gamma \frac{u}{|x|^2}-\epsilon u=|u|^{\frac{4}{N-2}}u &\hbox{in }\Omega u=0 & \hbox{on }\partial \Omega, \end{array}\right.
{
−
Δ
u
−
γ
∣
x
∣
2
u
​
−
ϵ
u
=
∣
u
∣
N
−
2
4
​
u
​
in Ω
u
=
0
​
onÂ
∂
Ω
,
​
when
ϵ
>
0
\epsilon>0
ϵ
>
0
is small and
γ
<
(
N
−
2
)
2
4
\gamma< {(N-2)^2\over4}
γ
<
4
(
N
−
2
)
2
​
. Setting
γ
j
=
(
N
−
2
)
2
4
(
1
−
j
(
N
−
2
+
j
)
N
−
1
)
∈
(
−
∞
,
0
]
\gamma_j= \frac{(N-2)^2}{4}\left(1-\frac{j(N-2+j)}{N-1}\right)\in(-\infty,0]
γ
j
​
=
4
(
N
−
2
)
2
​
(
1
−
N
−
1
j
(
N
−
2
+
j
)
​
)
∈
(
−
∞
,
0
]
for
j
∈
N
,
j \in \mathbb{N},
j
∈
N
,
we show that if
γ
≤
(
N
−
2
)
2
4
−
1
\gamma\leq \frac{(N-2)^2}{4}-1
γ
≤
4
(
N
−
2
)
2
​
−
1
and
γ
â‰
γ
j
\gamma \neq \gamma_j
γ
î€
=
γ
j
​
for any
j
j
j
, then for small
ϵ
\epsilon
ϵ
, the above equation has a positive --non variational-- solution that develops a bubble at the origin. If moreover
γ
<
(
N
−
2
)
2
4
−
4
,
\gamma<\frac{(N-2)^2}{4}-4,
γ
<
4
(
N
−
2
)
2
​
−
4
,
then for any integer
k
≥
2
k \geq 2
k
≥
2
, the equation has for small enough
ϵ
\epsilon
ϵ
, a sign-changing solution that develops into a superposition of
k
k
k
bubbles with alternating sign centered at the origin. The above result is optimal in the radial case, where the condition that
γ
â‰
γ
j
\gamma\neq \gamma_j
γ
î€
=
γ
j
​
is not necessary. Indeed, it is known that, if
γ
>
(
N
−
2
)
2
4
−
1
\gamma > \frac{(N-2)^2}{4}-1
γ
>
4
(
N
−
2
)
2
​
−
1
and
Ω
\Omega
Ω
is a ball
B
B
B
, then there is no radial positive solution for
ϵ
>
0
\epsilon>0
ϵ
>
0
small. We complete the picture here by showing that, if
γ
≥
(
N
−
2
)
2
4
−
4
\gamma\geq \frac{(N-2)^2}{4}-4
γ
≥
4
(
N
−
2
)
2
​
−
4
, then the above problem has no radial sign-changing solutions for
ϵ
>
0
\epsilon>0
ϵ
>
0
small. These results recover and improve what is known in the non-singular case, i.e., when
γ
=
0
\gamma=0
γ
=
0
.Comment: 41 pages, Updated version - if any - can be downloaded at http://www.birs.ca/~nassif
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Last time updated on 06/01/2019