Asymptotic behaviour of the Kazdan-Warner solution in the annulus ∗

Proof of Theorem 1.4 Let us multiply the equation (2.11) by u ′ p and integrate
from a to rp. We get
pu ′ rp
p − (N − 1)
a
rp
rp
− u
a
′′
1
2 u′ p (a)2 − (N − 1)
a
(u ′ p )2
r
(u ′ p )2
r
1 p+1
= up(rp)
p + 1
p+1 ||up|| ∞
=
p + 1
(2.23)
Using Proposition 2.4 and Theorem 2.2 we can pass to the limit in (2.23) and
we get, for N ≥ 3,
||up||
lim
p→∞
p+1
∞ 1
=
p + 1 2 ω′ (a) 2 r0
(ω
− (N − 1)
′ (r)) 2
r
= (N − 2) 2 2 N
2−N
From (2.24) we easily deduce that
2−N 2−N a + b N −1
2 N −2
(a 2−N − b 2−N ) 2
(p + 1)log ||up||∞ − log(p + 1) =
⎛
log
⎝(N − 2) 2 2 N
2−N
log ||up||∞ =
log(p + 1)
p + 1
2−N 2−N a + b N −1
2 N −2
a
(a 2−N − b 2−N ) 2
⎞
= lim
r→r −
1
2 0
ω′ (r) 2
⎠ + o(1) ⇒
(2.24)
log γ 1
+ + o , (2.25)
p + 1 p
and then the claim of Theorem 1.4 follows. ⊓⊔
We now recall some elementary fact about the Green’s function Ga,b(r, s) of
the operator
−u ′′ N − 1
− u
r
′ , r ∈ (a, b),
.
By definition, we have that for any smooth function f, we have that
b
v(r) = Ga,b(r, s)f(s)ds
is the solution of the problem
−u ′′ − N−1
r u′ = f in (a, b)
u(a) = u(b) = 0
a
It is not difficult to write down explicitly the function Ga,b(r, s). Indeed for
N ≥ 3 we have
s
Ga,b(r, s) =
N−1
(N − 2)(a2−N − b2−N ⎧
⎪⎨
2−N 2−N b − s
) ⎪⎩
r2−N − a2−N for a < r ≤ s
2−N 2−N s − a b2−N − r2−N for s < r < b
8