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

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procedure introduced in [AS] and also used in [AG]. Then, up to a suitable scaling, it is possible to associate to (1.1) the following limit problem defined in all R. − u ′′ = e u in R (1.11) A straightforward computation shows that this equation admits the following two-parameter solution Uα,y(r) = log 4α2 e √ 2(αr+y) (1 + e √ 2(αr+y) ) 2 for any α > 0 and y ∈ R. Note the similarities with equations involving critical Sobolev exponent or two-dimensional problems involving exponential nonlinearities. This analysis will be carried out in Section 3. In Section 4 we use the previous ”local” result to prove the nondegeneracy of the solution u for p large. We recall that the solution of problem (1.1) is nondegenerate if the following problem −∆v = pup−1v in Ω (1.12) v = 0 on ∂Ω, admits only the trivial solution v ≡ 0. We have the following theorem, Theorem 1.5. The solution u of problem (1.1) is nondegenerate provided p is large enough. The paper is organized as follows: in Section 2 we give the proof of Theorems 1.1, 1.3, 1.4 and Corollary 1.2. In Section 3 we discuss the blow-up argument near the maximum of up. In Section 4 we prove Theorem 1.5. 2 The ”global” convergence result Let us denote by H1 0,r(Ω) = {u ∈ H1 0(Ω) : u = u(|x|)}. Since the radial solution up to (1.1) is unique ([NN]), we can assume that it minimizes the following ratio We have the following inf u∈H 1 0,r (Ω) Ω Ω |∇u|2 up+1 2 p+1 = Proposition 2.1. For p > 1 let us denote by Ω |∇u|2 Then we have that Ip = inf u∈H 1 0,r (Ω) Ip ≤ Ω Ω |∇up| 2 Ω up+1 p up+1 2 p+1 ωN 4(N−2) a2−N −b2−N if N ≥ 3 1 8π log b−log a if N = 2 where ωN is the measure of the unit sphere in R N . 4 2 p+1 . (2.1) (2.2)
Proof. Let us consider the functions 2 ω(x) = a2−N − b2−N ⎧ ⎪⎨ ⎪⎩ a 2−N − |x| 2−N for a ≤ |x| ≤ r0 |x| 2−N − b 2−N for r0 ≤ |x| ≤ b and ⎧ ⎪⎨ log |x| − log a for a ≤ |x| ≤ r0 2 ω(x) = log b − log a ⎪⎩ log b − log |x| for r0 ≤ |x| ≤ b where r0 is given by ⎧ 1 2−N 2−N 2−N ⎪⎨ a +b 2 if N ≥ 3 r0 = ⎪⎩ √ ab if N = 2 for N ≥ 3 (2.3) for N = 2 (2.4) (2.5) Note that ω ∈ H1 0,r(Ω) and ||ω||∞ = 1. Recalling that lim p→∞ ||ω|| Lp (Ω) = ||ω|| L∞ (Ω) = 1 we get, for N ≥ 3, b inf u∈H 1 0,r (Ω) Ω 4(N − 2) Ω |∇u|2 ωN a2−N − b2−N up+1 2 p+1 Ω ≤ |∇ω|2 Ω ωp+1 2 p+1 → ωN a ω ′ (r) 2 r N−1 dr = (2.6) as p → ∞. This gives the claim in the case N ≥ 3. The case N = 2 is handled in the same way. ⊓⊔ Corollary 2.2. Let up the function which minimizes Ip. Then we have |∇up| 2 ≤ C and ≤ C. (2.7) Ω where C is a positive constant independent of p. Ω u p+1 p In the next lemma we recall a well known estimate, which implies that the maximum of the solution up is far away from zero. Lemma 2.3. We have that where λ1 denotes the first eigenvalue of −∆ in H 1 0 (Ω). ||up|| p−1 ∞ ≥ λ1 (2.8) Proof. Let us denote by e1 the first positive eigenfunction of −∆ in H1 0(Ω) and multiply (1.1) by e1. Integrating we have (2.9) λ1 upe1 = Ω Ω u p pe1 ≤ ||up|| p−1 ∞ upe1 Ω and the claim follows. ⊓⊔ 5
Page 1 and 2: Asymptotic behaviour of the Kazdan-
Page 3: (see Section 2 for some remarks on
Page 7 and 8: Lemma 2.7. Let ū the function defi
Page 9 and 10: and for N = 2, Ga,b(r, s) = s log b
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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