Local uniqueness and non-degeneracy of blow up solutions of mean field equations with singular data. (English) Zbl 1447.35134
The authors study the bubbling solutions of mean field equations with singular sources. The main result is the uniquness of bubbling solutions with respect to the blow up points under some non-degeneracy assumptions. The main tool of the proof is to apply the Pohozaev identity. This generalizes their previous work [J. Math. Pures Appl. (9) 123, 78–126 (2019; Zbl 1447.35152)] where the regular case was studied.
Reviewer: Zhijie Chen (Beijing)
MSC:
| 35J25 | Boundary value problems for second-order elliptic equations |
| 35J61 | Semilinear elliptic equations |
Citations:
Zbl 1447.35152References:
| [1] | Bartolucci, D., Global bifurcation analysis of mean field equations and the Onsager microcanonical description of two-dimensional turbulence, Calc. Var. Partial Differ. Equ., 58, 18 (2019) · Zbl 1417.35109 |
| [2] | Bartolucci, D.; Chen, C. C.; Lin, C. S.; Tarantello, G., Profile of blow up solutions to mean field equations with singular data, Commun. Partial Differ. Equ., 29, 7-8, 1241-1265 (2004) · Zbl 1062.35146 |
| [3] | Bartolucci, D.; De Marchis, F., On the Ambjorn-Olesen electroweak condensates, J. Math. Phys., 53, Article 073704 pp. (2012) · Zbl 1276.81124 |
| [4] | Bartolucci, D.; De Marchis, F., Supercritical mean field equations on convex domains and the Onsager’s statistical description of two-dimensional turbulence, Arch. Ration. Mech. Anal., 217, 2, 525-570 (2015) · Zbl 1317.35188 |
| [5] | Bartolucci, D.; De Marchis, F.; Malchiodi, A., Supercritical conformal metrics on surfaces with conical singularities, Int. Math. Res. Not., 24, 5625-5643 (2011) · Zbl 1254.30066 |
| [6] | Bartolucci, D.; Gui, C.; Hu, Y.; Jevnikar, A.; Yang, W., Mean field equation on torus: existence and uniqueness of evenly symmetric blow-up solutions, Discrete Contin. Dyn. Syst. (2019) |
| [7] | Bartolucci, D.; Gui, C.; Jevnikar, A.; Moradifam, A., A singular sphere covering inequality: uniqueness and symmetry of solutions to singular Liouville-type equations, Math. Ann., 374, 3-4, 1883-1922 (2019) · Zbl 1426.35127 |
| [8] | Bartolucci, D.; Jevnikar, A.; Lee, Y.; Yang, W., Uniqueness of bubbling solutions of mean field equations, J. Math. Pures Appl., 123, 78-126 (2019) · Zbl 1447.35152 |
| [9] | Bartolucci, D.; Jevnikar, A.; Lee, Y.; Yang, W., Non degeneracy, mean field equations and the Onsager theory of 2D turbulence, Arch. Ration. Mech. Anal., 230, 1, 397-426 (2018) · Zbl 1398.82043 |
| [10] | Bartolucci, D.; Jevnikar, A.; Lee, Y.; Yang, W., Local uniqueness of m-bubbling sequences for the Gel’fand equation, Commun. Partial Differ. Equ., 44, 6, 447-466 (2019) · Zbl 1414.35037 |
| [11] | Bartolucci, D.; Jevnikar, A.; Lin, C. S., Non-degeneracy and uniqueness of solutions to singular mean field equations on bounded domains, J. Differ. Equ., 266, 1, 716-741 (2019) · Zbl 1405.35046 |
| [12] | Bartolucci, D.; Lin, C. S., Uniqueness results for mean field equations with singular data, Commun. Partial Differ. Equ., 34, 7, 676-702 (2009) · Zbl 1185.35288 |
| [13] | Bartolucci, D.; Lin, C. S., Existence and uniqueness for mean field equations on multiply connected domains at the critical parameter, Math. Ann., 359, 1-44 (2014) · Zbl 1300.35058 |
| [14] | Bartolucci, D.; Lin, C. S.; Tarantello, G., Uniqueness and symmetry results for solutions of a mean field equation on \(\mathbb{S}^2\) via a new bubbling phenomenon, Commun. Pure Appl. Math., 64, 12, 1677-1730 (2011) · Zbl 1250.58009 |
| [15] | Bartolucci, D.; Malchiodi, A., An improved geometric inequality via vanishing moments, with applications to singular Liouville equations, Commun. Math. Phys., 322, 415-452 (2013) · Zbl 1276.58005 |
| [16] | Bartolucci, D.; Montefusco, E., Blow up analysis, existence and qualitative properties of solutions for the two dimensional Emden-Fowler equation with singular potential, Math. Methods Appl. Sci., 30, 18, 2309-2327 (2007) · Zbl 1149.35034 |
| [17] | Bartolucci, D.; Tarantello, G., Liouville type equations with singular data and their applications to periodic multivortices for the electroweak theory, Commun. Math. Phys., 229, 3-47 (2002) · Zbl 1009.58011 |
| [18] | Bartolucci, D.; Tarantello, G., Asymptotic blow-up analysis for singular Liouville type equations with applications, J. Differ. Equ., 262, 3887-3931 (2017) · Zbl 1364.35049 |
| [19] | Brezis, H.; Merle, F., Uniform estimates and blow-up behaviour for solutions of \(- \operatorname{\Delta} u = V(x) e^u\) in two dimensions, Commun. Partial Differ. Equ., 16, 8,9, 1223-1253 (1991) · Zbl 0746.35006 |
| [20] | Caglioti, E.; Lions, P. L.; Marchioro, C.; Pulvirenti, M., A special class of stationary flows for two dimensional Euler equations: a statistical mechanics description. II, Commun. Math. Phys., 174, 229-260 (1995) · Zbl 0840.76002 |
| [21] | Carlotto, A.; Malchiodi, A., Weighted barycentric sets and singular Liouville equations on compact surfaces, J. Funct. Anal., 262, 2, 409-450 (2012) · Zbl 1232.53035 |
| [22] | Chai, C. C.; Lin, C. S.; Wang, C. L., Mean field equations, hyperelliptic curves, and modular forms: I, Camb. J. Math., 3, 1-2, 127-274 (2015) · Zbl 1327.35116 |
| [23] | Chang, S. Y.A.; Chen, C. C.; Lin, C. S., Extremal functions for a mean field equation in two dimension, (Lecture on Partial Differential Equations. Lecture on Partial Differential Equations, New Stud. Adv. Math., vol. 2 (2003), Int. Press: Int. Press Somerville, MA), 61-93 · Zbl 1071.35040 |
| [24] | Chen, Z. J.; Kuo, T. J.; Lin, C. S., Hamiltonian system for the elliptic form of Painlevé VI equation, J. Math. Pures Appl., 106, 3, 546-581 (2016) · Zbl 1348.34141 |
| [25] | Chen, C. C.; Lin, C. S., Sharp estimates for solutions of multi-bubbles in compact Riemann surface, Commun. Pure Appl. Math., 55, 728-771 (2002) · Zbl 1040.53046 |
| [26] | Chen, C. C.; Lin, C. S., Topological degree for a mean field equation on Riemann surfaces, Commun. Pure Appl. Math., 56, 1667-1727 (2003) · Zbl 1032.58010 |
| [27] | Chen, C. C.; Lin, C. S., Mean field equations of Liouville type with singular data: sharper estimates, Discrete Contin. Dyn. Syst., 28, 3, 1237-1272 (2010) · Zbl 1211.35263 |
| [28] | Chen, C. C.; Lin, C. S., Mean field equation of Liouville type with singular data: topological degree, Commun. Pure Appl. Math., 68, 6, 887-947 (2015) · Zbl 1319.35283 |
| [29] | De Marchis, F., Generic multiplicity for a scalar field equation on compact surfaces, J. Funct. Anal., 259, 2165-2192 (2010) · Zbl 1211.58011 |
| [30] | Ding, W.; Jost, J.; Li, J.; Wang, G., Existence results for mean field equations, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 16, 653-666 (1999) · Zbl 0937.35055 |
| [31] | Djadli, Z., Existence result for the mean field problem on Riemann surfaces of all genuses, Commun. Contemp. Math., 10, 2, 205-220 (2008) · Zbl 1151.53035 |
| [32] | Esposito, P., Blowup solutions for a Liouville equation with singular data, SIAM J. Math. Anal., 36, 4, 1310-1345 (2005) · Zbl 1162.35350 |
| [33] | Esposito, P.; Grossi, M.; Pistoia, A., On the existence of blowing-up solutions for a mean field equation, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 22, 2, 227-257 (2005) · Zbl 1129.35376 |
| [34] | Gui, C.; Moradifam, A., The sphere covering inequality and its applications, Invent. Math., 214, 3, 1169-1204 (2018) · Zbl 1410.53037 |
| [35] | Gui, C.; Moradifam, A., Uniqueness of solutions of mean field equations in \(\mathbb{R}^2\), Proc. Am. Math. Soc., 146, 1231-1242 (2018) · Zbl 1387.35011 |
| [36] | Kazdan, J. L.; Warner, F. W., Curvature functions for compact 2-manifolds, Ann. Math., 99, 14-74 (1974) · Zbl 0273.53034 |
| [37] | Kowalczyk, M.; Musso, M.; del Pino, M., Singular limits in Liouville-type equations, Calc. Var. Partial Differ. Equ., 24, 1, 47-81 (2005) · Zbl 1088.35067 |
| [38] | Kuo, T. J.; Lin, C. S., Estimates of the mean field equations with integer singular sources: non-simple blow up, J. Differ. Geom., 103, 377-424 (2016) · Zbl 1358.35060 |
| [39] | Li, Y. Y., Harnack type inequality: the method of moving planes, Commun. Math. Phys., 200, 421-444 (1999) · Zbl 0928.35057 |
| [40] | Li, Y. Y.; Shafrir, I., Blow-up analysis for solutions of \(- \operatorname{\Delta} u = V(x) e^u\) in dimension two, Indiana Univ. Math. J., 43, 4, 1255-1270 (1994) · Zbl 0842.35011 |
| [41] | Lin, C. C., On the motion of vortices in two dimensions. I. Existence of the Kirchhoff-Routh function, Proc. Natl. Acad. Sci., 27, 570-575 (1941) · Zbl 0063.03560 |
| [42] | Lin, C. S., Uniqueness of solutions to the mean field equation for the spherical Onsager vortex, Arch. Ration. Mech. Anal., 153, 153-176 (2000) · Zbl 0968.35045 |
| [43] | Lin, C. S.; Lucia, M., Uniqueness of solutions for a mean field equation on torus, J. Differ. Equ., 229, 1, 172-185 (2006) · Zbl 1105.58005 |
| [44] | Lin, C. S.; Wang, C. L., Elliptic functions, Green functions and the mean field equations on tori, Ann. Math., 172, 2, 911-954 (2010) · Zbl 1207.35011 |
| [45] | Lin, C. S.; Yan, S., On the mean field type bubbling solutions for Chern-Simons-Higgs equation, Adv. Math., 338, 1141-1188 (2018) · Zbl 1404.35201 |
| [46] | Ma, L.; Wei, J., Convergence for a Liouville equation, Comment. Math. Helv., 76, 506-514 (2001) · Zbl 0987.35056 |
| [47] | Poliakovsky, A.; Tarantello, G., On a planar Liouville-type problem in the study of selfgravitating strings, J. Differ. Equ., 252, 3668-3693 (2012) · Zbl 1232.35167 |
| [48] | Prajapat, J.; Tarantello, G., On a class of elliptic problems in \(\mathbb{R}^2\): symmetry and uniqueness results, Proc. R. Soc. Edinb., Sect. A, 131, 967-985 (2001) · Zbl 1009.35018 |
| [49] | Spruck, J.; Yang, Y., On multivortices in the electroweak theory I: existence of periodic solutions, Commun. Math. Phys., 144, 1-16 (1992) · Zbl 0748.53059 |
| [50] | Suzuki, T., Global analysis for a two-dimensional elliptic eiqenvalue problem with the exponential nonlinearly, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 9, 4, 367-398 (1992) · Zbl 0785.35045 |
| [51] | Tarantello, G., Multiple condensate solutions for the Chern-Simons-Higgs theory, J. Math. Phys., 37, 3769-3796 (1996) · Zbl 0863.58081 |
| [52] | Troyanov, M., Prescribing curvature on compact surfaces with conical singularities, Trans. Am. Math. Soc., 324, 793-821 (1991) · Zbl 0724.53023 |
| [53] | Wei, J.; Zhang, L., Estimates for Liouville equation with quantized singularities (2019), Preprint |
| [54] | Wolansky, G., On steady distributions of self-attracting clusters under friction and fluctuations, Arch. Ration. Mech. Anal., 119, 355-391 (1992) · Zbl 0774.76069 |
| [55] | Wu, L.; Zhang, L., Uniqueness of bubbling solutions of mean field equations with non-quantized singularities (2019), Preprint |
| [56] | Yang, Y., Solitons in Field Theory and Nonlinear Analysis, Springer Monographs in Mathematics (2001), Springer: Springer New York · Zbl 0982.35003 |
| [57] | Zhang, L., Asymptotic behavior of blowup solutions for elliptic equations with exponential nonlinearity and singular data, Commun. Contemp. Math., 11, 395-411 (2009) · Zbl 1179.35137 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
