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Mean field equations on tori: Existence and uniqueness of evenly symmetric blow-up solutions
1. | Department of Mathematics, University of Rome "Tor Vergata", Via della ricerca scientifica n.1, 00133 Roma, Italy |
2. | Department of Mathematics, The University of Texas at San Antonio, San Antonio, TX 78249, USA |
3. | Scuola Normale Superiore, Piazza dei Cavalieri 3, 56126 Pisa, Italy |
4. | Division of Mathematics, Wuhan Institute of Physics and Mathematics, Chinese Academy of Science, Wuhan, Hubei 430071, China |
We are concerned with the blow-up analysis of mean field equations. It has been proven in [
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Uniqueness and symmetry results for solutions of a mean field equation on ${\mathbb{S}}^{2}$ via a new bubbling phenomenon, Comm. Pure Appl. Math., 64 (2011), 1677-1730.
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S. Baraket and F. Pacard,
Construction of singular limits for a semilinear elliptic equation in dimension $2$, Calc. Var. Partial Differential Equations, 6 (1998), 1-38.
doi: 10.1007/s005260050080. |
[2] |
D. Bartolucci, Global bifurcation analysis of mean field equations and the Onsager microcanonical description of two-dimensional turbulence, Calc. Var. Partial Differential Equations, 58 (2019), Art. 18, 37 pp.
doi: 10.1007/s00526-018-1445-4. |
[3] |
D. Bartolucci, C.-C. Chen, C.-S. Lin and G. Tarantello,
Profile of blow up solutions to mean field equations with singular data, Comm. Partial Differential Equations, 29 (2004), 1241-1265.
doi: 10.1081/PDE-200033739. |
[4] |
D. Bartolucci, C. F. Gui, A. Jevnikar and A. Moradifam,
A singular sphere covering inequality: Uniqueness and symmetry of solutions to singular Liouville-type equations, Math. Ann., 374 (2019), 1883-1922.
doi: 10.1007/s00208-018-1761-1. |
[5] |
D. Bartolucci, A. Jevnikar and C.-S. Lin,
Non-degeneracy and uniqueness of solutions to singular mean field equations on bounded domains, J. Diff. Eq., 266 (2019), 716-741.
doi: 10.1016/j.jde.2018.07.053. |
[6] |
D. Bartolucci, A. Jevnikar, Y. Lee and W. Yang,
Uniqueness of bubbling solutions of mean field equations, J. Math. Pures Appl., 123 (2019), 78-126.
doi: 10.1016/j.matpur.2018.12.002. |
[7] |
D. Bartolucci, A. Jevnikar, Y. Lee and W. Yang,
Non degeneracy, mean field equations and the Onsager theory of 2D turbulence, Arch. Rat. Mech. Anal., 230 (2018), 397-426.
doi: 10.1007/s00205-018-1248-y. |
[8] |
D. Bartolucci, A. Jevnikar, Y. Lee and W. Yang,
Local uniqueness of $m$-bubbling sequences for the Gel'fand equation, Comm. Partial Differential Equations, 44 (2019), 447-466.
doi: 10.1080/03605302.2019.1581801. |
[9] |
D. Bartolucci and F. De Marchis, On the Ambjorn-Olesen electroweak condensates, Jour. Math. Phys., 53 (2012), 073704, 15 pp.
doi: 10.1063/1.4731239. |
[10] |
D. Bartolucci and F. De Marchis,
Supercritical mean field equations on convex domains and the Onsager's statistical description of two-dimensional turbulence, Arch. Rat. Mech. Anal., 217 (2015), 525-570.
doi: 10.1007/s00205-014-0836-8. |
[11] |
D. Bartolucci, F. De Marchis and A. Malchiodi, Supercritical conformal metrics on surfaces with conical singularities, Int. Math. Res. Not. IMRN, (2011), 5625–5643.
doi: 10.1093/imrn/rnq285. |
[12] |
D. Bartolucci and C.-S. Lin,
Uniqueness results for mean field equations with singular data, Comm. in P. D. E., 34 (2009), 676-702.
doi: 10.1080/03605300902910089. |
[13] |
D. Bartolucci and C.-S. Lin,
Existence and uniqueness for mean field equations on multiply connected domains at the critical parameter, Math. Ann., 359 (2014), 1-44.
doi: 10.1007/s00208-013-0990-6. |
[14] |
D. Bartolucci, C.-S. Lin and G. Tarantello,
Uniqueness and symmetry results for solutions of a mean field equation on ${\mathbb{S}}^{2}$ via a new bubbling phenomenon, Comm. Pure Appl. Math., 64 (2011), 1677-1730.
doi: 10.1002/cpa.20385. |
[15] |
D. Bartolucci and A. Malchiodi,
An improved geometric inequality via vanishing moments, with applications to singular Liouville equations, Comm. Math. Phys., 322 (2013), 415-452.
doi: 10.1007/s00220-013-1731-0. |
[16] |
D. Bartolucci and G. Tarantello,
Liouville type equations with singular data and their applications to periodic multivortices for the electroweak theory, Comm. Math. Phys., 229 (2002), 3-47.
doi: 10.1007/s002200200664. |
[17] |
D. Bartolucci and G. Tarantello,
Asymptotic blow-up analysis for singular Liouville type equations with applications, J. Differential Equations, 262 (2017), 3887-3931.
doi: 10.1016/j.jde.2016.12.003. |
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L. Battaglia, M. Grossi and A. Pistoia, Non-uniqueness of blowing-up solutions to the Gelfand problem, Calculus of Variations and Partial Differential Equations, 58 (2019), arXiv: 1902.03484.
doi: 10.1007/s00526-019-1607-z. |
[19] |
H. Brezis and F. Merle,
Uniform estimates and blow-up behaviour for solutions of $-\Delta u = V(x)e^{u}$ in two dimensions, Comm. Partial Differential Equations, 16 (1991), 1223-1253.
doi: 10.1080/03605309108820797. |
[20] |
E. Caglioti, P.-L. Lions, C. Marchioro and M. Pulvirenti,
A special class of stationary flows for two-dimensional euler equations: A statistical mechanics description, Communications in Mathematical Physics, 143 (1992), 501-525.
doi: 10.1007/BF02099262. |
[21] |
E. Caglioti, P.-L. Lions, C. Marchioro and M. Pulvirenti,
A special class of stationary flows for two-dimensional euler equations: A statistical mechanics description. Ⅱ, Communications in Mathematical Physics, 174 (1995), 229-260.
doi: 10.1007/BF02099602. |
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D. Cao, S. L. Li and P. Luo,
Uniqueness of positive bound states with multi-bump for nonlinear Schrödinger equations, Calculus of Variations and Partial Differential Equations, 54 (2015), 4037-4063.
doi: 10.1007/s00526-015-0930-2. |
[23] |
D. Cao, E. S. Noussair and S. S. Yan,
Existence and uniqueness results on single peaked solutions of a semilinear problem, Annales de l'Institut Henri Poincaré, Analyse Non Linéaire, 15 (1998), 73-111.
doi: 10.1016/S0294-1449(99)80021-3. |
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Weighted barycentric sets and singular Liouville equations on compact surfaces, J. Funct. Anal., 262 (2012), 409-450.
doi: 10.1016/j.jfa.2011.09.012. |
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H. Chan, C.-C. Fu and C.-S. Lin,
Non-topological multi-vortex solutions to the self-dual Chern-Simons-Higgs equation, Communications in Mathematical Physics, 231 (2002), 189-221.
doi: 10.1007/s00220-002-0691-6. |
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S.-Y. A. Chang, C.-C. Chen and C.-S. Lin,
Extremal functions for a mean field equation in two dimension, Lecture on Partial Differential Equations, New Stud. Adv. Math., Int. Press, Somerville, MA, 2 (2003), 61-93.
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S. Chanillo and M. Kiessling,
Rotational symmetry of solutions of some nonlinear problems in statistical mechanics and in geometry, Communications in Mathematical Physics, 160 (1994), 217-238.
doi: 10.1007/BF02103274. |
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C.-C. Chen and C.-S. Lin,
Sharp estimates for solutions of multi-bubbles in compact riemann surfaces, Communications on Pure and Applied Mathematics, 55 (2002), 728-771.
doi: 10.1002/cpa.3014. |
[29] |
C.-C. Chen and C.-S. Lin,
Topological degree for a mean field equation on riemann surfaces, Communications on Pure and Applied Mathematics, 56 (2003), 1667-1727.
doi: 10.1002/cpa.10107. |
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C.-C. Chen and C.-S. Lin,
Mean field equation of liouville type with singular data: Topological degree, Communications on Pure and Applied Mathematics, 68 (2015), 887-947.
doi: 10.1002/cpa.21532. |
[31] |
C.-C. Chen and C.-S. Lin,
Mean field equations of Liouville type with singular data: Shaper estimates, Discrete Contin. Dyn. Syst., 28 (2010), 1237-1272.
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