Abstract
For an asymmetric sinh-Poisson problem arising as a mean field equation of equilibrium turbulence vortices with variable intensities of interest in hydrodynamic turbulence, we address the existence of bubbling solutions on compact Riemann surfaces. By using a Lyapunov–Schmidt reduction, we find sufficient conditions under which there exist bubbling solutions blowing up at m different points of S: positively at m1 points and negatively at m − m1 points with m ≥ 1 and m1 ∈ {0, 1,…,m}. Several examples in different situations illustrate our results in the sphere \(\mathbb{S}^{2}\) and flat two-torus \(\mathbb{T}\) including non-negative potentials with zero set non-empty.
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References
M. Ahmedou, T. Bartsch and T. Fiernkranz, Equilibria of vortex type Hamiltonians on closed surfaces, Topol. Methods Nonlinear Anal. 61 (2023), 239–256.
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.
D. Bartolucci and A. Pistoia, Existence and qualitative properties of concentrating solutions for the sinh-Poisson equation, IMA J. Appl. Math. 72 (2007), 706–729.
T. Bartsch, A. Pistoia and T. Weth. N-vortex equilibria for ideal fluids in bounded planar domains and new nodal solutions of the sinh-Poisson and the Lane–Emden–Fowler equations, Comm. Math. Phys. 297 (2010), 653–686.
L. Battaglia, A. Jevnikar, A. Malchiodi and D. Ruiz, A general existence result for the Toda system on compact surfaces, Adv. Math. 285 (2015), 937–979.
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, Comm. Math. Phys. 143 (1992), 501–525.
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, Comm. Math. Phys. 174 (1995), 229–260.
D. Chae and O. Imanuvilov, The existence of non-topological multivortex solutions in relativistic self-dual Chern–Simons theory, Comm. Math. Phys. 215 (2000) 119–142.
S-Y.A. Chang, C. C. Chen and C.-S. Lin, Extremal functions for a mean field equation in two dimension, in Lectures on Partial Differential Equations, International Press, Somerville, MA, 2003, pp. 61–93.
S.-Y. A. Chang, M. J. Gursky and P. C. Yang, The scalar curvature equation on 2- and 3-spheres, Calc. Var. Partial Differential Equations 1 (1993), 205–229.
S.-Y. A. Chang and P. C. Yang, Prescribing Gaussian curvature on S2, Acta Math. 159 (1987), 215–259.
S. Chanillo and M. Kiessling, Rotational symmetry of solutions of some nonlinear problems in statistical mechanics and in geometry, Comm. Math. Phys. 160 (1994), 217–238.
C. C. Chen and C. S. Lin, Sharp estimates for solutions of multi-bubbles in compact Riemann surfaces, Comm. Pure Appl. Math. 55 (2002), 728–771.
C. C. Chen and C. S. Lin, Topological degree for a mean field equation on Riemann surfaces, Comm. Pure Appl. Math. 56 (2003), 1667–1727.
C. C. Chen, C.-S. Lin, G. Wang, Concentration phenomena of two-vortex solutions in a Chern- Simons model, Ann. Sc. Norm. Sup. Pisa Cl. Sci. (5) 3 (2004), 367–397.
S.-S. Chern, An elementary proof of the existence of isothermal parameters on a surface, Proc. Amer. Math. Soc. 6 (1955), 771–782.
T. D’Aprile and P. Esposito, Equilibria of point-vortices on closed surfaces, Ann. Sc. Norm. Super. Pisa Cl. Sci. 17 (2017), 287–321.
Z. Djadli, Existence result for the mean field problem on Riemann surfaces of all genuses, Commun. Contemp. Math. 10 (2008), 205–220.
M. del Pino, P. Esposito, P. Figueroa and M. Musso, Non-topological condensates for the self-dual Chern–Simons–Higgs model, Comm. Pure Appl. Math. 68 (2015), 1191–1283.
M. del Pino, M. Kowalczykand M. Musso, Singular limits in Liouville-type equations, Cal. Var. Partial Differential Equations 24 (2005), 47–81.
M. del Pino, M. Musso and B. Ruf, New solutions for Trudinger–Moser critical equations in \(\mathbb{R}^{2}\), J. Funct. Anal. 258 (2010), 421–457.
W. Ding, J. Jost, J. Li and G. Wang, Existence results for mean field equations. Ann. Inst. H. Poincaré Anal. Non Lineairé 16 (1999), 653–666.
P. Esposito and P. Figueroa, Singular mean field equations on compact Riemann surfaces, Nonlinear Anal. 111 (2014), 33–65.
P. Esposito, P. Figueroa and A. Pistoia, On the mean field equation with variable intensities on pierced domains, Nonlinear Analysis 190 (2020), 111597.
P. Esposito, M. Grossi and A. Pistoia, On the existence of blowing-up solutions for a mean field equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005), 227–257.
P. Esposito, M. Musso and A. Pistoia, Concentrating solutions for a planar elliptic problem involving nonlinearities with large exponent, J. Differential Equations 227 (2006), 29–68.
P. Esposito and J. Wei, Non-simple blow-up solutions for the Neumann two-dimensional sinh-Gordon equation, Calc. Var. Partial Differential Equations 34 (2009), 341–375.
P. Figueroa, Singular limits for Liouville-type equations on the flat torus, Calc. Var. Partial Differential Equations 49 (2014), 613–647.
P. Figueroa, A note on sinh-Poisson equation with variable intensities on pierced domains, Asymptot. Anal. 122 (2021), 327–348.
P. Figueroa, Sign-changing bubble tower solutions for sinh-Poisson type equations on pierced domains, J. Differential Equations 367 (2023), 494–548.
P. Figueroa and M. Musso, Bubbling solutions for Moser–Trudinger type equations on compact Riemann surfaces, J. Funct. Anal. 275 (2018), 2684–2739.
P. Figueroa, L. Iturriaga and E. Topp, Sign-changing solutions for the sinh-Poisson equation with Robin Boundary condition, arXiv:2301.03688 [math.AP]
M. Grossi and A. Pistoia, Multiple blow-up phenomena for the sinh-Poisson equation, Arch. Rational Mech. Anal. 209 (2013), 287–320.
A. Jevnikar, An existence result for the mean-field equation on compact surfaces in a doubly supercritical regime, Proc. Royal Soc. Edinburgh A 143 (2013), 1021–1045.
A. Jevnikar, New existence results for the mean field equation on compact surfaces via degree theory, Rend. Semin. Mat. Univ. Padova 136 (2016), 11–17.
A. Jevnikar, Blow-up analysis and existence results in the supercritical case for an asymmetric mean field equation with variable intensities, J. Diff. Eq. 263 (2017), 972–1008
A. Jevnikar, J. Wei and W. Yang, On the Topological degree of the mean field equation with two parameters, Indiana Univ. Math. J. 67 (2018), 29–88.
A. Jevnikar, J. Wei and W. Yang, Classification of blow-up limits for sinh-Gordon equation, Differential Integral Equations 31 (2018), 657–684.
J. Jost, G. Wang, D. Ye and C. Zhou, The blow up analysis of solutions to the elliptic sinh-Gordon equation, Calc. Var. Partial Differential Equations 31 (2008), 263–276.
J. L. Kazdan and F. W. Warner, Curvature functions for compact 2-manifolds, Ann. of Math. (2) 99 (1974), 14–47.
M. K.-H. Kiessling, Statistical mechanics of classical particles with logarithmic interactions, Comm. Pure Appl. Math. 46 (1993), 27–56.
Y.Y. Li, On a singularly perturbed elliptic equation. Adv. Differential Equations 2 (1997), 955–980.
Y. Y. Li, Harnack type inequality: the method of moving planes, Comm. Math. Phys. 200 (1999), 421–444.
Y.-Y. Li and I. Shafrir, Blow-up analysis for solutions of − Δu = Veuin dimension two, Indiana Univ. Math. J. 43 (1994), 1255–1270.
C.-S. Lin and S. Yan, Existence of bubbling solutions for Chern–Simons model on a torus, Arch. Ration. Mech. Anal. 207 (2013), 353–392.
A. Malchiodi, Morse theory and a scalar field equation on compact surfaces, Adv. Differential Equations 13 (2008), 1109–1129.
M. Nolasco and G. Tarantello, Double vortex condensates in the Chern–Simons–Higgs theory, Cal. Var. Partial Differential Equations 9 (1999), 31–94.
H. Ohtsuka and T. Suzuki, Mean field equation for the equilibrium turbulence and a related functional inequality, Adv. Differential Equations 11 (2006), 281–304.
L. Onsager, Statistical hydrodynamics, Nuovo Cimento (9) 6 (1949), 279–287.
A. Pistoia and T. Ricciardi, Concentrating solutions for a Liouville type equation with variable intensities in 2D-turbulence, Nonlinearity 29 (2016), 271–297.
A. Pistoia and T. Ricciardi, Sign-changing tower of bubbles for a sinh-Poisson equation with asymmetric exponents, Discrete Contin. Dyn. Syst. 37 (2017), 5651–5692.
T. Ricciardi, Mountain-pass solutions for a mean field equation from two-dimensional turbulence, Differential Integral Equations 20 (2007), 561–575.
T. Ricciardi and R. Takahashi, Blow-up behavior for a degenerate elliptic sinh-Poisson equation with variable intensities, Calc. Var. Partial Differential Equations 55 (2016), Article no. 152.
T. Ricciardi, R. Takahashi, G. Zecca and X. Zhang, 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.
T. Ricciardi and G. Zecca, Minimal blow-upmasses and existence of solutions for an asymmetric sinh- Poisson equation, Math. Nachr. 290 (2017), 2375–2387
K. Sawada and T. Suzuki, Derivation of the equilibrium mean field equations of point vortex and vortex filament system, Theoret. Appl. Mech. Japan 56 (2008), 285–290.
G. Tarantello, Multiple condensate for Chern–Simons–Higgs theory, J. Math. Phys. 37 (1996), 3769–3796.
G. Tarantello, Selfdual Gauge Field Vortices, Birkhäuser, Boston, MA, 2008.
C. Q. Zhou, Existence of solution for mean field equation for the equilibrium turbulence, Nonlinear Anal. 69 (2008), 2541–2552.
C. Q. Zhou, Existence result for mean field equation of the equilibrium turbulence in the super critical case, Commun. Contemp. Math. 13 (2011), 659–673
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Author partially supported by grant Fondecyt Regular №1201884, Chile.
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Figueroa, P. Bubbling solutions for mean field equations with variable intensities on compact Riemann surfaces. JAMA 152, 507–555 (2024). https://doi.org/10.1007/s11854-023-0303-2
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DOI: https://doi.org/10.1007/s11854-023-0303-2