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A mean field equation involving positively supported probability measures: blow-up phenomena and variational aspects

Part of: Miscellaneous topics - Partial differential equations Elliptic equations and systems Partial differential equations

Published online by Cambridge University Press:  27 December 2018

Aleks Jevnikar  and
Wen Yang
Aleks Jevnikar
Affiliation:
Department of Mathematics, University of Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy (aleks.jevnikar@dm.unipi.it)
Wen Yang
Affiliation:
Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, P.O. Box 71010, Wuhan 430071, P. R. China (wyang@wipm.ac.cn)
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Abstract

We are concerned with an elliptic problem which describes a mean field equation of the equilibrium turbulence of vortices with variable intensities. In the first part of the paper, we describe the blow-up picture and highlight the differences from the standard mean field equation as we observe non-quantization phenomenon. In the second part, we discuss the Moser–Trudinger inequality in terms of the blow-up masses and get the existence of solutions in a non-coercive regime by means of a variational argument, which is based on some improved Moser–Trudinger inequalities.

Keywords

geometric PDEsmean field equationblow-up analysisvariational methods

MSC classification

Primary: 35J61: Semilinear elliptic equations
Secondary: 35J20: Variational methods for second-order elliptic equations 35R01: Partial differential equations on manifolds 35B44: Blow-up
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 2 , April 2019 , pp. 325 - 352
Copyright
Copyright © Royal Society of Edinburgh 2018 

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References

1Ao, W., Jevnikar, A. and Yang, W.. On the boundary behavior for the blow up solutions of the sinh-Gordon equation and rank N Toda systems in bounded domains. To appear in Int. Math. Res. Notices. doi:10.1093/imrn/rny263.Google Scholar
2Bahri, A. and Coron, J. M.. The scalar curvature problem on the standard three dimensional sphere. J. Funct. Anal. 95(1991), 106–172.Google Scholar
3Battaglia, L., Jevnikar, A., Malchiodi, A. and Ruiz, D.. A general existence result for the Toda system on compact surfaces. Adv. Math. 285 (2015), 937–979.Google Scholar
4Brezis, H. and Merle, F.. Uniform estimates and blow-up behavior for solutions of −Δu = V(x)e u in two dimensions. Comm. Partial Differ. Eq. 16 (1991), 1223–1254.Google Scholar
5Caglioti, E., Lions, P.L., Marchioro, C. and Pulvirenti, M.. A special class of stationary flows for two-dimensional Euler equations: a statistical mechanics description. Comm. Math. Phys. 143(1992), 501–525.Google Scholar
6Chang, S. Y. A. and Yang, P. C.. Prescribing Gaussian curvature on S 2. Acta Math. 159(1987), 215–259.Google Scholar
7Chang, S. Y. A., Gursky, M. J. and Yang, P. C.. The scalar curvature equation on 2- and 3- spheres. Calc. Var. and Partial Diff. Eq. 1(1993), 205–229.Google Scholar
8Chen, W. and Li, C.. Prescribing scalar curvature on S n. Pacific J. Math. 199(2001), 61–78.Google Scholar
9Chipot, M., Shafrir, I. and Wolansky, G.. On the solutions of Liouville systems. J. Differ. Eq. 140(1997), 59–105.Google Scholar
10Jevnikar, A.. An existence result for the mean field equation on compact surfaces in a doubly supercritical regime. Proc. Royal Soc. Edinb. Sect. A 143(2013), 1021–1045.Google Scholar
11Jevnikar, A.. New existence results for the mean field equation on compact surfaces via degree theory. Rend. Semin. Mat. Univ. Padova 136(2016), 11–17.Google Scholar
12Jevnikar, A.. A note on a multiplicity result for the mean field equation on compact surfaces. Adv. Nonlinear Stud. 16(2016), 221–229.Google Scholar
13Jevnikar, A. and Yang, W.. Analytic aspects of the Tzitzéica equation: blow-up analysis and existence results. Calc. Var. and Partial Diff. Eq. 56(2017), 56: 43.Google Scholar
14Jevnikar, A., Wei, J. and Yang, W.. Classification of blow-up limits for the sinh-Gordon equation. Differential and Integral Equations 31(2018), 657–684.Google Scholar
15Jevnikar, A.. Blow-up analysis and existence results in the supercritical case for an asymmetric mean field equation with variable intensities. J. Differ. Eq. 263(2017), 972–1008.Google Scholar
16Jevnikar, A., Wei, J. and Yang, W.. On the Topological degree of the Mean field equation with two parameters. Indiana Univ. Math. J. 67(2018), 29–88.Google Scholar
17Kiessling, M. K. H.. Statistical mechanics of classical particles with logarithmic interactions. Comm. Pure Appl. Math. 46(1993), 27–56.Google Scholar
18Li, Y. Y.. Prescribing scalar curvature on S n and related problems. I. J. Differ. Eq. 120(1995), 319–410.Google Scholar
19Li, Y. Y.. Harnack type inequality: the method of moving planes. Comm. Math. Phys. 200(1999), 421–444.Google Scholar
20Li, Y. Y. and Shafrir, I.. Blow-up analysis for solutions of −Δu = V e u in dimension two. Indiana Univ. Math. J. 43(1994), 1255–1270.Google Scholar
21Lin, C. S. and Zhang, L.. A topological degree counting for some Liouville systems of mean field type. Comm. Pure Appl. Math. 64(2011), 556–590.Google Scholar
22Lin, C. S. and Zhang, L.. Energy concentration for Singular Toda systems with B 2 and G 2 types of Cartan matrices. Int. Math. Res. Notices. (2016), 5076–5105.Google Scholar
23Lin, C. S., Wei, J. C. and Zhang, L.. Classifcation of blowup limits for SU(3) singular Toda systems. Anal. PDE 8(2015), 807–837.Google Scholar
24Malchiodi, A.. Topological methods for an elliptic equation with exponential nonlinearities. Discrete Contin. Dyn. Syst. 21(2008), 277–294.Google Scholar
25Malchiodi, A. and Ndiaye, C. B.. Some existence results for the Toda system on closed surfaces. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 18(2007), 391–412.Google Scholar
26Ohtsuka, H. and Suzuki, T.. Mean field equation for the equilibrium turbulence and a related functional inequality. Adv. Differ. Equ. 11 (2006a), 281–304.Google Scholar
27Ohtsuka, H. and Suzuki, T.. A blowup analysis of the mean field equation for arbitrarily signed vortices. Self-similar solutions of nonlinear PDE 74 (2006b), 185–197.Google Scholar
28Ohtsuka, H., Ricciardi, T. and Suzuki, T.. Blow-up analysis for an elliptic equation describing stationary vortex flows with variable intensities in 2D-turbulence. J. Differ. Eq. 249 (2010), 1436–1465.Google Scholar
29Onsager, L.. Statistical hydrodynamics. Nuovo Cimento Suppl. 6 (1949), 279–287.Google Scholar
30Poliakovsky, A. and Tarantello, G.. On a planar Liouville-type problem in the study of selfgravitating strings. J. Differ. Eq. 252(2012), 3668–3693.Google Scholar
31Pistoia, A. and Ricciardi, T.. Concentrating solutions for a Liouville type equation with variable intensities in 2D-turbulence. Nonlinearity 29(2016), 271–297.Google Scholar
32Ricciardi, T. and Zecca, G.. Blow-up analysis for some mean field equations involving probability measures from statistical hydrodynamics. Differ. Integral Equ. 25(2012), 201–222.Google Scholar
33Ricciardi, T., Takahashi, R., Zecca, G. and Zhang, X.. On the existence and blow-up of solutions for a mean field equation with variable intensities. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 27(2016), 413–429.Google Scholar
34Sawada, K. and Suzuki, T.. Derivation of the equilibrium mean field equations of point vortex and vortex filament system. Theoret. Appl. Mech. Japan 56 (2008), 285–290.Google Scholar
35Schoen, R. and Zhang, D.. Prescribed scalar curvature on the n-sphere. Calc. Var. 4(1996), 1–25.Google Scholar
36Struwe, M.. The existence of surfaces of constant mean curvature with free boundaries. Acta Math. 160 (1988), 19–64.Google Scholar
37Tarantello, G.. Analytical, geometrical and topological aspects of a class of mean field equations on surfaces. Discrete Contin. Dyn. Syst. 28(2010), 931–973.Google Scholar
38Tarantello, G.. Blow up analysis for a cosmic strings equation. J. Funct. Anal. 272(2017), 255–338.Google Scholar
39Zhou, C.. Existence result for mean field equation of the equilibrium turbulence in the super critical case. Commun. Contemp. Math. 13(2011), 659–673.Google Scholar