CONCENTRATING SOLUTIONS FOR A PLANAR ... - CAPDE

Proof. We have already shown that the map ξ → φ ξ is a C 1 −map into
H 1 0 (Ω) and then, F (ξ) is a C1 -function of ξ.
Since D ξ F (ξ) = 0, we have that
∫
0 = −
= −
= −
Ω
2∑ m∑
i=1 j=1
2∑ m∑
i=1 j=1
(∆(U ξ + φ ξ ) + (U ξ + φ ξ ) p ) (D ξ U ξ + D ξ φ ξ )
∫
c ij (ξ) e U j
Z ij (D ξ U ξ + D ξ φ ξ )
Ω
∫
c ij (ξ) e U j
Z ij D ξ U ξ +
Ω
2∑
m∑
i=1 j=1
∫
c ij (ξ) D ξ (e U j
Z ij )φ ξ
Ω
since ∫ Ω eU j
Z ij φ ξ = 0. By the expression of U ξ , we have that:
m∑
[
1
∂ (ξj ) i
U ξ = −
2
P
p−1
s=1 γµ s
2
+
p 2 (p − 1) w 1 + 1 p ∇w 0 · y + 1 p 2 ∇w 1 · y
(
1
+
2
P
p−1
γδ j µ j
(
1
=
2
P
p−1
γδ j µ
j
2
p − 1 U δ s ,ξ s
− 2Z 0s +
) ∣∣y=
x−ξs
δs
Z ij − 1 p ∂ iw 0 ( x − ξ j
) − 1 δ j p 2 ∂ iw 1 ( x − ξ j
)
δ j
Z ij − 1 p ∂ iw 0 ( x − ξ j
) − 1 δ j p 2 ∂ iw 1 ( x − ξ j
)
δ j
29
(
2
p(p − 1) w 0 (5.3)
]
∂ (ξj ) i
log µ s
)
)
+ O( 1 γ ),
since P : L ∞ (Ω) → L ∞ 1
(Ω) is a continuous operator (apply to
p−1 U δ s ,ξ s
,
Z 0s , 1 p w j( x−ξs
δ s
), ∇w j ( x−ξs
δ s
) · ( x−ξs
δ s
) for j = 0, 1 which are bounded in Ω in
view of Lemma 2.1), and
∂ (ξj ) i
(
e U h
Z lh
)
(
= −4δ h e U δ
h
il
δh 2 + |x − ξ h| 2 − 6(x − ξ )
h) l (x − ξ j ) i
(δh 2 + |x − ξ h| 2 ) 2 δ hj
+3e U h
Z 0h Z lh ∂ (ξj ) i
log µ h , (5.4)
where δ hj denotes the Kronecker’s symbol. Hence, by (5.3)-(5.4) for i = 1, 2
and j = 1, . . . , m we get that
1
2∑ m∑
)
0 = ∂ (ξj ) i
F (ξ) = −
2
c lh (ξ)
(P Z ij , P Z lh
p−1
H
γδ j µ l=1 h=1
1 0 (Ω)
j
( )
1
+O + ‖φ ξ ‖ ∞ |∂
pγδ (ξj ) i
(e
j
∫Ω
U ∑ 2 m∑
h
Z lh )| |c lh (ξ)|
l=1 h=1