CONCENTRATING SOLUTIONS FOR A PLANAR ... - CAPDE

|x − ξ j | ≥ ε for any j = 1, . . . , m,
⎛
⎞−1 m∑
∣
δ (
⎝
j ⎠
(δj 2 + |x − ξ ∆U
j| 2 ) 3 ξ + U p ξ
2
j=1
)
(x) ∣ (
≤ Ce p 4 ( C )
p )p + pe − p 2
13
= O(pe − p 4 ), (2.21)
and, if |x − ξ i | ≤ ε for some i = 1, . . . , m,
|∆U ξ + U p ξ | = ∣ (
1
8
γδi 2 2
−
µ
p−1 (1 + |y| 2 ) 2 + 1 p ˜f 0 (y) + 1 p ˜f 2 1 (y) (2.22)
i
− 1 8
p (1 + |y| 2 ) 2 w 0(y) − 1 )
8
p 2 (1 + |y| 2 ) 2 w 1(y)
+U p ξ (δ iy + ξ i ) + O(pe − p 2 )
∣ ∣ ,
where we denote y = x−ξ i
δ i
. By (2.14) we deduce that, for x = δ i y + ξ i
(
U p ξ (x) = ( p
2
) p 1 + 1
p−1 p v ∞(y) + 1 p
γµ
2 w 0(y) + 1 p p ) p
p 3 w 1(y) + O( e− 4
p |y| + e− 4
p ) .
Since (
i
p
2
p−1
γµ i
U p ξ (x) = 1
) p =
1
2 , by (2.10) we get for |x − ξ i | ≤ ε √ δ i
γδi 2µ p−1
i
(
γδ 2 i µ 2
p−1
i
8
(1 + |y| 2 ) 2 [1 + 1 p
w 0 (y) − 1 )
8
2 log2 (
(1 + |y| 2 ) 2 )
+ 1 8
(w
p 2 1 − log(
(1 + |y| 2 ) 2 )w 0 + 1 8
3 log3 (
(1 + |y| 2 ) 2 )
+ w2 0
2 + 1 8
8 log4 (
(1 + |y| 2 ) 2 ) − w )
0 8
2 log2 (
(1 + |y| 2 ) 2 )
(
log 6 )]
(|y| + 2)
+O
p 3 + p 2 e − p 4 y + p 2 e − p 4 , y = x − ξ i
.
δ i
Hence, in this region we obtain that
⎛
⎞−1 m∑
∣
δ (
⎝
j ⎠
(δj 2 + |x − ξ ∆U
j| 2 ) 3 ξ + U p ξ
2
j=1
≤ ∣ (δ2 i + |x − ξ i| 2 ) 3 2
(∆U ξ + U p )
ξ
(x) ∣ δ i
)
(x) ∣ ∣ (2.23)
≤ C (
γ (1 + |y|2 ) 3 1 log 6 )
(|y| + 2)
2 O
p 3 (1 + |y| 2 ) 2 ≤ C p 4 , y = x − ξ i
.
δ i
On the other hand, if ε √ δ i ≤ |x − ξ i | ≤ ε we have that
( p
)
U p e
ξ (x) = O 2 1
γ (1 + |y| 2 ) 2 , y = x − ξ i
,
δ i