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

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Note that, by Theorem 1.4 and Proposition 3.2 we get up(εpr + rp) p−1 = e (p−1)(log up(εpr+rp)−||up||∞+||up||∞) = (4.11) log p γ 1 (p−1)(log up(εpr+rp)−||up||∞+1+ p + p +o( e p)) = since up(εpr + rp) − ||up||∞ = ũp(r) 1 + o p p p γ p (p−1)(ũp(r)+log p + p +o(log p )) = e e ũp(r) pe γ (1 + o(1)), (4.12) with ũp and γ as in Theorem 1.4 and Proposition 3.2 respectively. Using now the definition of εp and Theorem 1.4 we derive that pε 2 pup(εpr + rp) p−1 → e v(r) 4e = √ 2r (1 + e √ 2r ) 2 (4.13) because ||up||p ∞ p → eγ again by Theorem 1.4. Since vp is bounded, by Ascoli-Arzelá Theorem, we get that vp → v in (R) and by (4.10), (4.11), we deduce that v satisfies C 1 loc − v ′′ = 4e √ 2r (1 + e √ 2r ) 2 v in R. (4.14) By Lemma 4.2 the claim follows. ⊓⊔ Proof of Theorem 1.5 By Lemma 4.1 we can suppose that the point where v achieves its maximum coincides with rp (otherwise we multiply (4.1) by −1). Let us suppose that there exists a sequence pn → ∞ and a solution vpn ≡ vn ≡ 0 of the problem ⎧ ⎪⎨ −v ⎪⎩ ′′ N−1 n − r v′ pn−1 n = pnu vn in Ω pn vn(rpn) = 1 (4.15) vn = 0 on ∂Ω, Let us set upn ≡ un, ε 2 pn = 1 pn||un|| pn−1 ∞ and ||un||∞ = un(rn). Finally let us set ˜vn(r) = vn(εpnr + rn). By Proposition 4.3 we have that there exist real constant α and β such that ˜vn(r) → v(r) = α 1 − e√ 2r √2r1 − e + β √ 2r + 2 Step 1: β = 0 in (4.16). By (4.16) we get that and 1 + e √ 2r lim r→±∞ ˜vn(r) = 1 + e √ 2r in C 1 loc (R) (4.16) lim r→0 ˜vn(r) = 2β (4.17) ⎧ ⎪⎨ +∞ if β < 0 ⎪⎩ −∞ if β > 0 12 (4.18)
From (4.17) and (4.18) we derive that there exist R1 < 0 < R2 such that ˜vn(R1) = ˜vn(R2) = 0. Then, coming back to the function vn we get that vn(εnR1 + rpn) = vn(εnR2 + rpn) = 0. But we would get the existence of three nodal region to the function vn and this is impossible. Step 2: A contradiction arises. By the previous steps we get that (4.16) becomes ˜vn(r) → α 1 − e√ 2r 1 + e √ 2r in C 1 loc (R) (4.19) From (4.19) we get that vn(0) → 0. On the other hand, since vn(rn) = 1 a contradiction arises. Then vn ≡ 0 for n large enough. ⊓⊔ References [AG] Adimurthi and M. Grossi, Asymptotic estimates for a two-dimensional problem with polynomial nonlinearity, Proc. Amer. Math. Soc., 132, (2004), 1013-1019. [AS] Adimurthi and M. Struwe, Global compactness properties of semilinear elliptic equations with critical exponential growth, J. Funct. Anal., 175 (2000), 125-167. [BM] L. Boccardo and F. Murat, Increase of power leads to bilateral problems, Composite media and homogenization theory, Dal Maso and Dell’Antonio editors, (1995), World Scientific. [EG] K. El Mehdi and M. Grossi, Asymptotic estimates and qualitative properties of an elliptic problem in dimension two, Adv. Nonlinear Stud., 4, (2004), 15-36. [GNN] B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), 209-243. [KW] J.L. Kazdan and F.W. Warner, Remarks on some quasilinear elliptic equations, Comm. Pure Appl. Math., 28, (1975), 567-597. [NN] W.M. Ni and R. Nussbaum, Uniqueness and nonuniqueness for positive radial solutions of ∆u + f(u, r) = 0,Comm. Pure Appl. Math., 38 (1985), 67-108. [Pa] P. Padilla, Symmetry properties of positive solutions of elliptic equations on symmetric domains, Appl. Anal., 64 (1997), 153-169. [P] S. Pohozaev, Eigenfunctions of the equation ∆u + λf(u) = 0, Soviet. Math. Dokl. 6 (1965), 1408-1411. [RW1] X. Ren and J. Wei, On a two-dimensional elliptic problem with large exponent in nonlinearity, Trans. Amer. Math. Soc., 343, (1994), 749-763. [RW2] X. Ren and J. Wei, Single-point condensation and least-energy solutions, Proc. Amer. Math. Soc., 124, (1996), 111-120. 13
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 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: Let us write down the equation sati

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

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