CONCENTRATING SOLUTIONS FOR A PLANAR ... - CAPDE

21
since ∫ Ω |P u j| = O(| log δ j |) = O(p) and
∫
∫
δ j
1
B(ξ j ,ε) (δj 2 + |x − ξ |u
j| 2 ) 3 j | ≤
|u
2 R 2 (1 + |y| 2 ) 3 j |(δ j y + ξ j ) = O(p).
2
Finally, by (3.11)
∫
R j φ = O
Ω
(∫
Ω
e U j
(|x − ξ j | + e − p 4 )
)
= O(e − p 4 ). (3.17)
Hence, inserting (3.14)-(3.17) in (3.13) we obtain that
16π
3 C j = o(1)
for any j = 1, . . . , m. Necessarily, C j = 0 and the claim is proved.
5 th Step. We establish the validity of the a priori estimate:
‖φ‖ ∞ ≤ Cp‖h‖ ∗ (3.18)
for solutions of problem (3.3)-(3.5) and h ∈ C 0,α (¯Ω).
gives
⎛
⎞
2∑ m∑
‖φ‖ ∞ ≤ Cp ⎝‖h‖ ∗ + |c ij | ⎠
since
i=1 j=1
‖e U j
Z ij ‖ ∗ ≤ 2‖e U j
‖ ∗ ≤ 16.
The previous Step
Hence, arguing by contradiction of (3.18), we can proceed as in Step 3 and
suppose further that
2∑ m∑
p n ‖h n ‖ ∗ → 0 , p n |c n ij| ≥ δ > 0 as n → +∞.
i=1 j=1
We omit the dependence on n. It suffices to estimate the values of the
constants c ij . For i = 1, 2 and j = 1, . . . , m, multiply (3.3) by P Z ij and,
integrating by parts, get:
2∑
m∑
l=1 h=1
∫
( )
c lh P Zlh , P Z ij + hP Z
H 1 ij =
0 Ω
−
∫
Ω
∫
Ω
W (x)φP Z ij (3.19)
e U j
Z ij φ,
since ∆P Z ij = ∆Z ij = −e U j
Z ij . For i = 1, 2 and j = 1, . . . , m we have the
following expansions:
( )
∂H
P Z ij = Z ij − 8πδ j ∂(ξ j ) i
(·, ξ j ) + O δj
3 ( ) (3.20)
P Z 0j = Z 0j − 1 + O
δ 2 j