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

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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 of of Corollary 1.2 It is enough to compare directly the function ω in Theorem 1.1 with the function Ga,b(r, r0). ⊓⊔ 3 The ”local” convergence result Let us begin this section with the following classification result, Proposition 3.1. All the solution of the problem are given by with α and y real constants. − z ′′ = e z in R (3.1) Uα,y(r) = log 4α2 e √ 2(αr+y) (1 + e √ 2(αr+y) ) 2 Proof. It is a straightforward computation. ⊓⊔ Let us consider the function with up(rp) = ||up||∞ and pε 2 p p ũp(r) = (up(εpr + rp) − ||up||∞) (3.2) ||up||∞ p−1 ||up|| ∞ = 1. We have that ũp solves the problem ⎧ ⎪⎨ −ũ ′′ p − N−1 εpr+rp εpũ ′ p = 1 + ũp p p ⎪⎩ ũp(0) = ũ ′ p (0) = 0, in a−rp In the next proposition we study the limit of the function ũp. Proposition 3.2. We have that √ 2r 4e where U(r) = log (1+e √ 2r ) 2 εp b−rp , εp (3.3) ũp → U uniformly in C 1 loc(R) (3.4) is the only solution of the problem. ⎧ ⎪⎨ −z ′′ = ez in R ⎪⎩ z(0) = z ′ (0) = 0 9 (3.5)
Page 1 and 2: Asymptotic behaviour of the Kazdan-
Page 3 and 4: (see Section 2 for some remarks on
Page 5 and 6: Proof. Let us consider the function
Page 7: Lemma 2.7. Let ū the function defi
Page 11 and 12: Let us write down the equation sati
Page 13: From (4.17) and (4.18) we derive th

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

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