CONCENTRATING SOLUTIONS FOR A PLANAR ... - CAPDE

24 PIERPAOLO ESPOSITO, MONICA MUSSO, AND ANGELA PISTOIA
for some constant C j , j = 1, . . . , m. Hence, in (3.25) we have a better
estimate since by Lebesgue theorem the term
∫
(
32y i
(1 + |y| 2 ) 3 w 0 − v ∞ − 1 )
2 v2 ∞ (y) ˆφ j (y)
converges to
B(0, √ ε )
δ j
∫
32y i (|y| 2 (
− 1)
C j
(1 + |y| 2 ) 4 w 0 − v ∞ − 1 )
2 v2 ∞ (y) = 0.
R 2
Therefore, we get that the R.H.S. in (3.19) satisfies: R.H.S. = o( 1 p
), and in
turn, ∑ 2 ∑ mh=1
l=1 |c lh | = O(‖h‖ ∗ ) + o( 1 p
). This contradicts
and the claim is established.
2∑ m∑
p |c ij | ≥ δ > 0,
i=1 j=1
6 th Step. We prove the solvability of (3.3)–(3.5). To this purpose, we
consider the spaces:
and
2∑ m∑
K ξ = { c ij P Z ij : c ij ∈ R for i = 1, 2, j = 1, . . . , m}
i=1 j=1
∫
Kξ ⊥ = {φ ∈ L 2 (Ω) :
Let Π ξ : L 2 (Ω) → K ξ defined as:
Ω
e U j
Z ij φ = 0 for i = 1, 2, j = 1, . . . , m}.
2∑ m∑
Π ξ φ = c ij P Z ij ,
i=1 j=1
where c ij are uniquely determined (as it follows by (3.22)-(3.23)) by the
system:
∫
Ω
e U h
Z lh
⎛
⎝φ −
2∑
m∑
i=1 j=1
c ij P Z ij
⎞
⎠ = 0 for any l = 1, 2 , h = 1, . . . , m.
Let Π ⊥ ξ = Id − Π ξ : L 2 (Ω) → Kξ ⊥ . Problem (3.3)–(3.5), expressed in a weak
form, is equivalent to find φ ∈ Kξ
⊥ ∩ H1 0 (Ω) such that
∫
(φ, ψ) H 1
0 (Ω) = (W φ − h) ψ dx, for all ψ ∈ Kξ ⊥ ∩ H0 1 (Ω).
Ω
With the aid of Riesz’s representation theorem, this equation gets rewritten
in Kξ
⊥ ∩ H1 0 (Ω) in the operatorial form
(Id − K)φ = ˜h, (3.27)