CONCENTRATING SOLUTIONS FOR A PLANAR ... - CAPDE

16 PIERPAOLO ESPOSITO, MONICA MUSSO, AND ANGELA PISTOIA
Given h ∈ C(¯Ω), we consider the linear problem of finding a function
φ ∈ W 2,2 (Ω) such that
2∑ m∑
L(φ) = h + c ij e U j
Z ij , in Ω, (3.3)
i=1 j=1
∫
φ = 0, on ∂Ω, (3.4)
e U j
Z ij φ = 0, for all i = 1, 2, j = 1, . . . , m, (3.5)
Ω
for some coefficients c ij , i = 1, 2 and j = 1, . . . , m. Here and in the sequel,
for any i = 0, 1, 2 and j = 1, . . . , m we denote
⎧
⎪⎨
Z ij (x) := z i ( x − ξ j
δ j
) =
The main result of this Section is the following:
⎪⎩
|x−ξ j | 2 −δ 2 j
δ 2 j +|x−ξ j| 2 if i = 0
4δ j (x−ξ j ) i
δ 2 j +|x−ξ j| 2 if i = 1, 2.
Proposition 3.1. Let ε > 0 be fixed. There exist p 0 > 0 and C > 0 such
that, for h ∈ C(¯Ω) there is a unique solution to problem (3.3)-(3.5), for any
p > p 0 and ξ ∈ O ε , which satisfies
‖φ‖ ∞ ≤ Cp‖h‖ ∗ . (3.6)
Proof. The proof of this result consists of six steps.
1 st Step. The operator L satisfies the maximum principle in ˜Ω := Ω \
∪ m j=1 B(ξ j, Rδ j ) for R large, independent on p. Namely,
if L(ψ) ≤ 0 in ˜Ω and ψ ≥ 0 on ∂ ˜Ω, then ψ ≥ 0 in ˜Ω.
In order to prove this fact, we show the existence of a positive function Z
in ˜Ω satisfying L(Z) < 0. We define Z to be
( )
m∑ a(x − ξj )
Z(x) = z 0 , a > 0.
δ j
j=1
First, observe that, if |x−ξ j | ≥ Rδ j for R > 1 a
, then Z(x) > 0. On the other
hand, we have:
⎛
m∑
W (x)Z(x) ≤ D 0 ⎝
e U j(x)
j=1
⎞
⎠ Z(x) ≤ D 0 Z(x)
m∑
j=1
8δ 2 j
|x − ξ j | 4 ,
where D 0 is the constant in Lemma 3.1. Further,by definition of z 0 ,
m∑
−∆Z(x) = a 2 8δ2 j (a2 |x − ξ j | 2 − δj 2)
(a 2 |x − ξ j | 2 + δj 2)3
≥ 1 3
j=1
m∑
j=1
8a 2 δ 2 j
(a 2 |x − ξ j | 2 + δ 2 j )2 ≥ 4 27
m∑
j=1
8δ 2 j
a 2 |x − ξ j | 4