CONCENTRATING SOLUTIONS FOR A PLANAR ... - CAPDE

The parameters µ j will be chosen later. Observe that for any j ≠ i and
x = δ i y + ξ i
1
[P U δj ,ξ j
(x) + 1 (
p P w 0 ( x − ξ )
j
) + 1 (
δ j p 2 P w 1 ( x − ξ )]
j
)
δ j
γµ
2
p−1
j
= 8π
γµ
2
p−1
j
[
G(ξ i , ξ j ) 1 − C 0
4p − C ]
1
4p 2
p + O( e− 4 + |x − ξ i |
).
γ
Hence, we get that the function U ξ (x) is a good approximation for a solution
to problem (1.1) exhibiting m points of concentration provided
log(8µ 4 i ) =
(
8πH(ξ i , ξ i ) 1 − C 0
4p − C )
1
4p 2 + log δ (
i
C 0 + C )
1
p p
+8π ∑ 2
p−1 [
µ i
2
G(ξ j , ξ i ) 1 − C 0
p−1
4p − C ]
1
4p
j≠i µ
.
A direct computation shows that, for p large, µ satisfies
j=1
γµ
2
p−1
j
j
∑
µ i = e − 3 2πH(ξ
4 e i ,ξ i )+2π
j≠i G(ξ j,ξ i ) 1 (1 + O( )). (2.13)
p
Indeed, with this choice of the parameters µ i , we have that
m∑ 1
[P U δj ,ξ j
(x) + 1 (
p P w 0 ( x − ξ )
j
) + 1 (
δ j p 2 P
= 1
γµ
2
p−1
i
for x = δ i y + ξ i .
w 1 ( x − ξ j
δ j
)
(p + v ∞ (y) + 1 p w 0(y) + 1 p 2 w 1(y) + O(e − p 4 |y| + e
− p 4 )
)
(2.14)
Remark 2.1. Let us remark that U ξ is a positive function. Since |v ∞ +
1
p w 0 + 1 w
p 2 1 | ≤ C in |y| ≤ ɛ δ i
, by (2.14) we get that U ξ is positive in B(ξ i , ɛ)
for any i = 1, . . . , m. Moreover, by (2.11) we get that
(
P w i ( x − ξ )
j
) → −2πC i G(·, ξ j )
δ j
in C 1 -norm on |x − ξ j | ≥ ɛ, i = 0, 1, and then
P U δj ,ξ j
+ 1 (
p P w 0 ( x − ξ )
j
) + 1 (
δ j p 2 P w 1 ( x − ξ )
j
)
δ j
→ 8πG(·, ξ j )
in C 1 -norm on |x−ξ j | ≥ ɛ. Hence, since ∂G
∂ n (·, ξ j) < 0 on ∂Ω, U ξ is a positive
function in Ω.
)]
11