CONCENTRATING SOLUTIONS FOR A PLANAR ... - CAPDE

14 PIERPAOLO ESPOSITO, MONICA MUSSO, AND ANGELA PISTOIA
since (1 + s p )p ≤ e s . Thus, in this region
⎛
⎞−1 m∑
∣
δ ( )
⎝
j ⎠
j=1 (δj 2 + |x − ξ ∆U
j| 2 ) 3 ξ + U p ξ
(x) ∣ (2.24)
2
(
)
p
= O
≤ Cpe − p x − ξ i
(1 + |y| 2 ) 1 8 , y = .
2
δ i
By (2.21), (2.23) and (2.24) we obtain the desired result.
3. Analysis of the linearized operator
In this Section, we prove bounded invertibility of the operator L, uniformly
on ξ ∈ O ε , by using L ∞ -norms introduced in (2.18). Let us recall that
L(φ) = ∆φ + W ξ φ, where W ξ (x) = pU p−1
ξ
(x). For simplicity of notation, we
will omit the dependence of W ξ on ξ.
As in Proposition 2.1, we have for the potential W (x) the following expansions.
If |x − ξ i | ≤ ε for some i = 1, . . . , m
p
W (x) = p(
γµ
2
p−1
i
+O( e− p 4
p |y| + e− p 4
p ) ) p−1
= δ −2
i
(
) p−1 (1 + 1 p v ∞(y) + 1 p 2 w 0(y) + 1 p 3 w 1(y) (3.1)
1 + 1 p v ∞(y) + 1 p 2 w 0(y) + 1 p 3 w 1(y) + O( e− p 4
p |y| + e− p 4
p ) ) p−1
where again we use the notation y = x−ξ i
δ i
. In this region, we have that
W (x) ≤ C δi
2 e v∞(y) e − 1 p v∞(y) ( )
= O e U i(x)
,
since v ∞ (y) ≥ −2p. Indeed, by Taylor expansions of exponential and logarithmic
functions as in (2.10), we obtain that, if |x − ξ i | ≤ ε √ δ i (and
|y| ≤ √ ε
δi
),
W (x) = δ −2
i
=
(
1 + 1 p v ∞(y) + 1 p 2 w 0(y) + 1 p 3 w 1(y) + O( e− p 4
p |y| + e− p 4
p ) ) p−1
(
8
δi 2(1 + |y|2 ) 2 1 + 1 p (w 0 − v ∞ − 1 2 v2 ∞) + O( log4 (|y| + 2)
p 2 )
If |x − ξ i | ≥ ε for any i = 1, . . . , m:
Summing up, we have that
W (x) = O(p( C p )p−1 ).
)
.
□