CONCENTRATING SOLUTIONS FOR A PLANAR ... - CAPDE

We expand the term ∫ Ω |∇U ξ| 2 : in view of (2.14) and (2.20) we have that
∫
m∑
∫ (
|∇U ξ | 2 1
=
2
e U j
− 1
Ω
p−1
j=1
B(ξ
γµ
j ,ε) pδ 2 ∆w 0 ( x − ξ j
)
j δ j
j
− 1
p 2 δj
2 ∆w 1 ( x − ξ )
j
) + O(p 2 e − p 2 ) U ξ + O(e − p 2 )
δ j
m∑
∫ (
1
8
=
4
j=1 γ 2 p−1 B(0,
µ
ε ) (1 + |y| 2 ) 2 − 1 p ∆w 0 − 1 )
p 2 ∆w 1 + O(p 2 e −p ) ×
δ
j
j
×
(p + v ∞ + 1 p w 0 + 1 )
p 2 w 1 + O(e − p 4 |y| + e
− p 4 ) + O(e − p 2 )
=
m∑
j=1
1
γ 2 µ
= 8πmp
γ 2
since µ − 4
p−1
j
get that:
4
p−1
j
− 32π
γ 2
( ∫
8πp +
m ∑
j=1
R 2 (
8
(1 + |y| 2 ) 2 v ∞ − ∆w 0 ) + O( 1 )
p )
log µ j + m 8
γ
∫R 2 (
2 (1 + |y| 2 ) 2 v ∞ − ∆w 0 ) + O( 1 p 3 )
= 1 − 4 p log µ j + O( 1 p 2 ). Recalling property (2.13) of µ i , then we
∫
Ω
|∇U ξ | 2 = 8πmp
γ 2
31
− 64π2
γ 2 ϕ m(ξ 1 , . . . , ξ m ) + 24πm
γ 2 (5.6)
+ m 8
γ
∫R 2 (
2 (1 + |y| 2 ) 2 v ∞ − ∆w 0 ) + O( 1 p 3 )
uniformly for ξ ∈ O ε . Hence, using (4.6) and (5.6) we get that
∫
∇U ξ ∇φ ξ + 1 ∫
|∇φ ξ | 2 = O( 1 ). (5.7)
Ω 2 Ω
Finally, inserting (5.6)-(5.7) in (5.5), we get that
uniformly for ξ ∈ O ε .
F (ξ) = 4πmp
γ 2 − 32π2
γ 2 ϕ m(ξ 1 , . . . , ξ m ) + 4πm
γ 2
+ m ∫
8
2γ 2 (1 + |y| 2 ) 2 v ∞ − ∆w 0 ) + O( 1 p 3 )
R 2 (
Now, we want to show that the expansion of F (ξ) in terms of ϕ m (ξ) holds
in a C 1 -sense:
Lemma 5.3. Let ε > 0. The following expansion holds:
∇ (ξj ) i
F (ξ) = − 32π2
γ 2 ∇ (ξ j ) i
ϕ m (ξ 1 , . . . , ξ m ) + o( 1 p 2 )
uniformly for ξ ∈ O ε , for any j = 1, . . . , m and i = 1, 2.
p 7 2