Asymptotic analysis on positive solutions of the Lane-Emden system with nearly critical exponents
Authors: Kim Seunghyeok , Moon Sang-Hyuck 
Published: January 2023

We concern a family

{
(

u

ε

,

v

ε

)

}

ε
>
0

\{(u_{\varepsilon },v_{\varepsilon })\}_{\varepsilon > 0}

of solutions of the Lane-Emden system on a smooth bounded convex domain

Ω
\Omega

in

R

N

\mathbb {R}^N

\[
<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 Subscript epsilon Baseline equals v Subscript epsilon Superscript p Baseline 2nd Column a m p semicolon in normal upper Omega comma 2nd Row 1st Column minus normal upper Delta v Subscript epsilon Baseline equals u Subscript epsilon Superscript q Super Subscript epsilon Baseline 2nd Column a m p semicolon in normal upper Omega comma 3rd Row 1st Column u Subscript epsilon Baseline comma v Subscript epsilon Baseline greater-than 0 2nd Column a m p semicolon in normal upper Omega comma 4th Row 1st Column u Subscript epsilon Baseline equals v Subscript epsilon Baseline equals 0 2nd Column a m p semicolon on partial-differential normal upper Omega comma EndLayout">

{

−
Δ

u

ε

=

v

ε

p

a
m
p
;
in 
Ω
,

−
Δ

v

ε

=

u

ε

q

ε

a
m
p
;
in 
Ω
,

u

ε

,

v

ε

>
0

a
m
p
;
in 
Ω
,

u

ε

=

v

ε

=
0

a
m
p
;
on 
∂
Ω
,

\begin {cases} -\Delta u_{\varepsilon } = v_{\varepsilon }^p & \text {in } \Omega , \\ -\Delta v_{\varepsilon } = u_{\varepsilon }^{q_{\varepsilon }} & \text {in } \Omega , \\ u_{\varepsilon },\, v_{\varepsilon } > 0 & \text {in } \Omega , \\ u_{\varepsilon } = v_{\varepsilon } =0 & \text {on } \partial \Omega , \end {cases}

\]
for

N
≥
4

N \ge 4

,

max
{
1
,

3

N
−
2

}
>
p
>

q

ε

\max \{1,\frac {3}{N-2}\} > p > q_{\varepsilon }

and small
\[

ε
≔

N

p
+
1

+

N

q

ε

+
1

−
(
N
−
2
)
>
0.

\varepsilon ≔\frac {N}{p+1} + \frac {N}{q_{\varepsilon }+1} - (N-2) > 0.

\]
This system appears as the extremal equation of the Sobolev embedding
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W Superscript 2 comma left-parenthesis p plus 1 right-parenthesis slash p Baseline left-parenthesis normal upper Omega right-parenthesis right-arrow with hook upper L Superscript q Super Subscript epsilon Superscript plus 1 Baseline left-parenthesis normal upper Omega right-parenthesis">

W

2
,
(
p
+
1
)

/

p

(
Ω
)
↪

L

q

ε

+
1

(
Ω
)

W^{2,(p+1)/p}(\Omega ) \hookrightarrow L^{q_{\varepsilon }+1}(\Omega )

, and is also closely related to the Calderón-Zygmund estimate. Under the natural energy condition, we prove that the multiple bubbling phenomena may arise for the family

{
(

u

ε

,

v

ε

)

}

ε
>
0

\{(u_{\varepsilon },v_{\varepsilon })\}_{\varepsilon > 0}

, and establish a detailed qualitative and quantitative description. If

p
>

N

N
−
2

p > \frac {N}{N-2}

, the nonlinear structure of the system makes the interaction between bubbles so strong, so the determination process of the blow-up rates and locations is completely different from that of the classical Lane-Emden equation. If

p
≥

N

N
−
2

p \ge \frac {N}{N-2}

, the blow-up scenario is relatively close to that of the classical Lane-Emden equation, and only single-bubble solutions can exist. Even in the latter case, we have to devise a new method to cover all

p
p

near

N

N
−
2

\frac {N}{N-2}

. We also deduce a general existence theorem that holds on any smooth bounded domains.