CONCENTRATING SOLUTIONS FOR A PLANAR ... - CAPDE

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where δ lj denotes the Kronecker’s symbol, and hence, we get that
|
2∑
k=1 l=1
m∑
∫
c kl
Ω
≤ C‖φ ξ ‖ ∞ max |c kl |
k , l
[∂ (ξj ) i
(e U l
Z kl
)
+ η(x − ξ j )∂ xi
(e U l
Z kl
)]
φ ξ |
∫
Ω
(
)
e U l
+ O(e − 3 4 p ) = O( 1 p 7 )
in view of |Z 0l Z kl | ≤ 2, (4.5)-(4.6). Inserting in (5.8), we get that
∫
Ω
(
∆u ξ + u p )
ξ
∂ (ξj ) i
φ ξ = −
2∑
k=1 l=1
+O( 1 p 7 ).
m∑
∫
c kl
Ω
e U l
Z kl ∂ xi (η(x − ξ j )φ ξ )(5.9)
Since pφ ξ → 0 in C 1 -norm away from ξ 1 , . . . , ξ m , (5.9) gives that
∫
Ω
( )
∆u ξ + u p ξ
∂ (ξj ) i
φ ξ = −
= −
2∑
k=1 l=1
∫
m∑
∫
c kl
B(ξ j ,ɛ)
always in view of (4.1) and (4.5)-(4.6).
B(ξ j ,ɛ)
(
∆u ξ + u p ξ
e U l
Z kl ∂ xi φ ξ + O( 1 p 7 )
)
∂ xi φ ξ + O( 1 p 7 ) (5.10)
Now, by Lemma 2.1 we get that | 1 δ s
∂ xi w l ( x−ξs
δ s
)| ≤ C for any l = 1, 2 and
for x away from ξ s . Hence, by the expression of U ξ and (2.2), (2.11) we get
that for |x − ξ j | ≤ 2ɛ:
m∑
(
1
∂ xi U ξ =
2
∂ xi P U δs ,ξ s
+ 1 (
p−1
p ∂ x i
P w 0 ( x − ξ )
s
)
(5.11)
δ
s=1 γµ
s
s
+ 1 (
p 2 ∂ x i
P w 1 ( x − ξ ))
s
)
δ s
m∑
(
1
=
2
∂ xi U δs,ξs + 1 ∂ xi w 0 ( x − ξ s
) + 1
p−1
pδ
s=1 γµ
s δ s p 2 ∂ xi w 1 ( x − ξ )
s
) + O( 1 δ s δ s γ )
s
(
1
= −
2
Z ij − 1
p−1 p ∂ x i
w 0 ( x − ξ j
) − 1 δ
γδ j µ
j p 2 ∂ x i
w 1 ( x − ξ )
j
) + O( 1 δ j γ ),
j
since ∂ xi U δs ,ξ s
= − 1 δ s
Z is . Since, as already observed, ∂ xi w l ( x−ξ j
δ j
) = O(δ j )
uniformly away from ξ j , l = 1, 2, by (5.11) in particular we get:
∂ xi U ξ = O( 1 ), (5.12)
γ