×

zbMATH — the first resource for mathematics

Sharp estimates for solutions of multi-bubbles in compact Riemann surfaces. (English) Zbl 1040.53046
The authors claim that the Leray-Schauder degree \(d(\rho)\) of the nonlinear elliptic equation \[ \Delta_0 u+\rho\Biggl({h(x)e^u\over h(x) e^ud\mu}- 1\Biggr)= 0,\tag{1} \] on a compact Riemann surface \((M,ds)\), where \(\Delta_0\) is the Beltrami-Laplace operator, is \(1\) for \(0< \rho< 8\pi\) and \[ {1\over m!} (-\chi(M)+ 1)\cdots(-\chi(M)+ m)\quad\text{for }8m\pi< \rho< 8(m+ 1)\pi, \] where \(\chi(M)\) is the Euler characteristic class of \(M\) [C. C. Chen and C. S. Lin, ibid. 56, 1667–1727 (2003; Zbl 1032.58010)]. To prove the claim, the estimate \[ \rho_k- 8m\pi= {2\over m} \sum^m_{j=1} h-1(p_{k,j})(\Delta_0 \log h(p_{k,j})+ 8m\pi- 2K(p_{k,j}))\lambda_{k,j} e^{-\lambda_{k,j}}+ O(e^{-\lambda_{k,j}}),\tag{2} \] where \(\lim_{k\to\infty}\rho_k= 8m\pi\), \(p_{k,j}\) are centers of the bubbles of \(u_k\) and \(\lambda_{k,j}\) are the local maxima of \(u_k\), is proved in this paper (Theorem 1.1).
Equation (1) is the Euler-Lagrange equation of the nonlinear functional \[ J_\phi(\phi)= {1\over 2}\int_M| \nabla\phi|^2d\mu- \rho\log\Biggl(\int_M he^\phi d\mu\Biggr), \] and arises as the Nirenberg or Kazdan-Warner problem in geometry, from self-dual condensate solutions of some Chern-Simons-Higgs model in physics, and so on.
In Section 2, previous studies on (1) are reviewed. The main results in this section are cited from [Y. Y. Li, Commun. Math. Phys. 200, 421–444 (1999; Zbl 0928.35057)]. Then, estimating solutions \(u_k\) for \(\rho_k\) and \[ \begin{aligned} \eta_{k,j}(x) &= u_k(x)- v_{k,j}(x)- (G^*_j(x)- G^*_j (p_{k,j})),\\ v_{k,j}(x) &= \log\Biggl({e^{\lambda_{k,j}}\over (1+ (\rho_k h_k(p_{k,j})/8) e^{\lambda_{k,j}}| x-q_{k,j}|^2)^2}\Biggr),\end{aligned} \] where \(\lambda_{k,j}= u_k(p_{k,j})\), \(\nabla v_{k,j}(p_{k,j})= \nabla\log\widehat h(p_{k,j})\) and \(\lambda_{k,j}= u_k(p_{k,j})= \max_{\overline B_{\delta_0(p)}} u_k(x)\to +\infty\), \(G^*_j(x)= \rho_{k,j}(\widetilde G_j(x)+ \sum_{i\neq j}\rho_{k,j}G(x, p_{k,j})\), \(\widetilde G_j(x)= G(x,p_{k,j})+ {1\over 2\pi}\log| x|\), where \(G(x,p)\) is the Green function of \(\Delta_0\) with singularity at \(p\), (2) is proved in Section 3. Some estimates are proved in Section 4. A sharper estimate of \(\eta_{k,j}(x)\) used in the calculation of the Leray-Schauder degree is given in Section 5 (Theorem 5.1). In Section 6 similar estimates for the Dirichlet problem of (1) on \(\mathbb{R}^2\) are given (Theorem 6.1–6.3).

MSC:
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
35J60 Nonlinear elliptic equations
58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
58E12 Variational problems concerning minimal surfaces (problems in two independent variables)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Brezis, J Funct Anal 115 pp 344– (1993) · Zbl 0794.35048 · doi:10.1006/jfan.1993.1094
[2] Brezis, Comm Partial Differential Equations 16 pp 1223– (1991) · Zbl 0746.35006 · doi:10.1080/03605309108820797
[3] Caffarelli, Comm Math Phys 168 pp 321– (1995) · Zbl 0846.58063 · doi:10.1007/BF02101552
[4] Caglioti, Comm Math Phys 143 pp 501– (1992) · Zbl 0745.76001 · doi:10.1007/BF02099262
[5] Caglioti, Comm Math Phys 174 pp 229– (1995) · Zbl 0840.76002 · doi:10.1007/BF02099602
[6] Chanillo, Comm Math Phys 160 pp 217– (1994) · Zbl 0821.35044 · doi:10.1007/BF02103274
[7] Chanillo, Geom Funct Anal 5 pp 924– (1995) · Zbl 0858.35035 · doi:10.1007/BF01902215
[8] Chang, Calc Var Partial Differential Equations 1 pp 205– (1993) · Zbl 0822.35043 · doi:10.1007/BF01191617
[9] Chang, Acta Math 159 pp 215– (1987) · Zbl 0636.53053 · doi:10.1007/BF02392560
[10] Chen, Comm Anal Geom 6 pp 1– (1998) · Zbl 0903.35009 · doi:10.4310/CAG.1998.v6.n1.a1
[11] Chen, Ann Inst H Poincaré Anal Non Linéaire 18 pp 271– (2001) · Zbl 0995.35004 · doi:10.1016/S0294-1449(00)00060-3
[12] ; Singular limits of a nonlinear eigenvalue problem in two dimensions. Preprint.
[13] ; Topological degree for a mean field equation on Riemann surfaces. Preprint. · Zbl 1032.58010
[14] Chen, Duke Math J 63 pp 615– (1991) · Zbl 0768.35025 · doi:10.1215/S0012-7094-91-06325-8
[15] Cheng, Calc Var Partial Differential Equations 11 pp 203– (2000) · Zbl 0996.53008 · doi:10.1007/s005260000037
[16] Cheng, Adv Differential Equations 5 pp 1253– (2000)
[17] Chipot, J Differential Equations 140 pp 59– (1997) · Zbl 0902.35039 · doi:10.1006/jdeq.1997.3316
[18] Ding, Asian J Math 1 pp 230– (1997) · Zbl 0955.58010 · doi:10.4310/AJM.1997.v1.n2.a3
[19] Ding, Calc Var Partial Differential Equations 7 pp 87– (1998) · Zbl 0928.58021 · doi:10.1007/s005260050100
[20] Ding, Ann Inst H Poincaré Anal Non Linéaire 16 pp 653– (1999) · Zbl 0937.35055 · doi:10.1016/S0294-1449(99)80031-6
[21] Ding, Comment Math Helv 74 pp 118– (1999) · Zbl 0913.53032 · doi:10.1007/s000140050079
[22] Kazdan, Ann of Math (2) 99 pp 14– (1974) · Zbl 0273.53034 · doi:10.2307/1971012
[23] Kiessling, Comm Pure Appl Math 46 pp 27– (1993) · Zbl 0811.76002 · doi:10.1002/cpa.3160460103
[24] Li, Comm Math Phys 200 pp 421– (1999) · Zbl 0928.35057 · doi:10.1007/s002200050536
[25] Li, Indiana Univ Math J 43 pp 1255– (1994) · Zbl 0842.35011 · doi:10.1512/iumj.1994.43.43054
[26] Lin, Duke Math J 104 pp 501– (2000) · Zbl 0964.35038 · doi:10.1215/S0012-7094-00-10437-1
[27] Lin, Calc Var Partial Differential Equations 10 pp 291– (2000) · Zbl 0996.53007 · doi:10.1007/s005269900026
[28] Lin, Arch Ration Mech Anal 153 pp 153– (2000) · Zbl 0968.35045 · doi:10.1007/s002050000085
[29] On a nonlinear problem in differential geometry. Dynamical systems (Proc Sympos, Univ Bahia, Salvador, 1971), 273-280. Edited by Academic, New York, 1973. · doi:10.1016/B978-0-12-550350-1.50026-6
[30] Nagasaki, Asymptotic Anal 3 pp 173– (1990)
[31] Nolasco, Arch Ration Mech Anal 145 pp 161– (1998) · Zbl 0980.46022 · doi:10.1007/s002050050127
[32] Nolasco, Calc Var Partial Differential Equations 9 pp 31– (1999) · Zbl 0951.58030 · doi:10.1007/s005260050132
[33] Nolasco, Comm Math Phys 213 pp 599– (2000) · Zbl 0998.81047 · doi:10.1007/s002200000252
[34] Ricciardi, Differential Integral Equations 11 pp 745– (1998)
[35] Ricciardi, Comm Pure Appl Math 53 pp 811– (2000) · Zbl 1029.35207 · doi:10.1002/(SICI)1097-0312(200007)53:7<811::AID-CPA2>3.0.CO;2-F
[36] Spruck, Ann Inst H Poincaré Anal Non Linéaire 12 pp 75– (1995) · Zbl 0836.35007 · doi:10.1016/S0294-1449(16)30168-8
[37] Struwe, Boll Unione Mat Ital Sez B Artic Ric Mat 8 pp 109– (1998)
[38] Suzuki, Ann Inst H Poincaré Anal Non Linéaire 9 pp 367– (1992) · Zbl 0785.35045 · doi:10.1016/S0294-1449(16)30232-3
[39] Tarantello, J Math Phys 37 pp 3769– (1996) · Zbl 0863.58081 · doi:10.1063/1.531601
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.