doi: 10.3934/dcds.2020039

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

Received  February 2019 Revised  April 2019 Published  October 2019

Fund Project: D. Bartolucci is partially supported by MIUR Excellence Department Project awarded to the Department of Mathematics, Univ. of Rome Tor Vergata, CUP E83C18000100006; C.Gui and Y. Hu are partially supported by NSF grant DMS-1601885; W. Yang is partially supported by NSFC No.11801550

We are concerned with the blow-up analysis of mean field equations. It has been proven in [6] that solutions blowing-up at the same non-degenerate blow-up set are unique. On the other hand, the authors in [18] show that solutions with a degenerate blow-up set are in general non-unique. In this paper we first prove that evenly symmetric solutions on an arbitrary flat torus with a degenerate two-point blow-up set are unique. In the second part of the paper we complete the analysis by proving the existence of such blow-up solutions using a Lyapunov-Schmidt reduction method. Moreover, we deduce that all evenly symmetric blow-up solutions come from one-point blow-up solutions of the mean field equation on a "half" torus.

Citation: Daniele Bartolucci, Changfeng Gui, Yeyao Hu, Aleks Jevnikar, Wen Yang. Mean field equations on tori: Existence and uniqueness of evenly symmetric blow-up solutions. Discrete & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2020039
References:
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References:
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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.  Google Scholar

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[5]

D. BartolucciA. 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.  Google Scholar

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D. BartolucciA. JevnikarY. 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.  Google Scholar

[7]

D. BartolucciA. JevnikarY. 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.  Google Scholar

[8]

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[9]

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[10]

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[11]

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[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.  Google Scholar

[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.  Google Scholar

[14]

D. BartolucciC.-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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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.  Google Scholar

[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.  Google Scholar

[20]

E. CagliotiP.-L. LionsC. 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.  Google Scholar

[21]

E. CagliotiP.-L. LionsC. 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.  Google Scholar

[22]

D. CaoS. 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.  Google Scholar

[23]

D. CaoE. 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.  Google Scholar

[24]

A. Carlotto and A. Malchiodi, 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.  Google Scholar

[25]

H. ChanC.-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.  Google Scholar

[26]

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