# American Institute of Mathematical Sciences

November  2017, 37(11): 5651-5692. doi: 10.3934/dcds.2017245

## Sign-changing tower of bubbles for a sinh-Poisson equation with asymmetric exponents

 1 Dipartimento SBAI, Università di Roma "La Sapienza", Via Antonio Scarpa 16, 00161 Rome, Italy 2 Dipartimento di Matematica e Applicazioni "R. Caccioppoli", Università di Napoli Federico Ⅱ, Via Cintia, 80126 Naples, Italy

Received  March 2017 Revised  May 2017 Published  July 2017

Motivated by the statistical mechanics description of stationary 2D-turbulence, for a sinh-Poisson type equation with asymmetric nonlinearity, we construct a concentrating solution sequence in the form of a tower of singular Liouville bubbles, each of which has a different degeneracy exponent. The asymmetry parameter $γ∈(0, 1]$ corresponds to the ratio between the intensity of the negatively rotating vortices and the intensity of the positively rotating vortices. Our solutions correspond to a superposition of highly concentrated vortex configurations of alternating orientation; they extend in a nontrivial way some known results for $\gamma=1$. Thus, by analyzing the case $\gamma≠1$ we emphasize specific properties of the physically relevant parameter $\gamma$ in the vortex concentration phenomena.

Citation: Angela Pistoia, Tonia Ricciardi. Sign-changing tower of bubbles for a sinh-Poisson equation with asymmetric exponents. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5651-5692. doi: 10.3934/dcds.2017245
##### References:
 [1] 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, Commun. Math. Phys., 174 (1995), 229-260. doi: 10.1007/BF02099602. Google Scholar [2] M. del Pino, P. Esposito and M. Musso, Nondegeneracy of entire solutions of a singular Liouville equation, Proc. Am. Math. Soc., 140 (2012), 581-588. doi: 10.1090/S0002-9939-2011-11134-1. Google Scholar [3] M. del Pino, M. Kowalczyk and M. Musso, Singular limits in Liouville-type equations, Calc. Var. Partial Differential Equations, 24 (2005), 47-81. doi: 10.1007/s00526-004-0314-5. Google Scholar [4] P. Esposito, M. Grossi and A. Pistoia, On the existence of blowing-up solutions for a mean field equation, Ann. I. H. Poincaré, 22 (2005), 227-257. doi: 10.1016/j.anihpc.2004.12.001. Google Scholar [5] G. L. Eyink and K. R. Sreenivasan, Onsager and the theory of hydrodynamic turbulence, Reviews of Modern Physics, 78 (2006), 87-135. doi: 10.1103/RevModPhys.78.87. Google Scholar [6] M. Grossi, C. Grumiau and F. Pacella, Lane Emden problems with large exponents and singular Liouville equations, J. Math. Pures Appl., 101 (2014), 735-754. doi: 10.1016/j.matpur.2013.06.011. Google Scholar [7] M. Grossi and A. Pistoia, Multiple blow-up phenomena for the sinh-Poisson equation, Arch. Rational Mech. Anal., 209 (2013), 287-320. doi: 10.1007/s00205-013-0625-9. Google Scholar [8] A. Jevnikar, An existence result for the mean-field equation on compact surfaces in a doubly supercritical regime, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 1021-1045. doi: 10.1017/S030821051200042X. Google Scholar [9] A. Jevnikar and W. Yang, Analytic aspects of the Tzitzéica equation: Blow-up analysis and existence results, Calc. Var. Partial Differential Equations, 56 (2017), Paper No. 43, 38 pp. doi: 10.1007/s00526-017-1136-6. Google Scholar [10] D. D. Joseph and T. S. Lundgren, Quasilinear problems driven by positive sources, Arch. Rat. Mech. Anal., 49 (1973), 241-269. doi: 10.1007/BF00250508. Google Scholar [11] J. Jost, G. Wang, D. Ye and C. Zhou, The blow up analysis of solutions of the elliptic sinh-Gordon equation, Calc. Var. Partial Differential Equations, 31 (2008), 263-276. doi: 10.1007/s00526-007-0116-7. Google Scholar [12] G. Joyce and D. Montgomery, Negative temperature states for the two-dimensional guiding centre plasma, J. Plasma Phys., 10 (1973), 107-121. Google Scholar [13] C. S. Lin, An expository survey on recent development of mean field equations, Discr. Cont. Dynamical Systems, 19 (2007), 387-410. doi: 10.3934/dcds.2007.19.387. Google Scholar [14] A. Malchiodi, Topological methods for an elliptic equation with exponential nonlinearities, Discr. Cont. Dynamical Systems, 21 (2008), 277-294. doi: 10.3934/dcds.2008.21.277. Google Scholar [15] J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J, 20 (1970/71), 1077-1092. doi: 10.1512/iumj.1971.20.20101. Google Scholar [16] C. Neri, Statistical Mechanics of the $N$-point vortex system with random intesities on a bounded domain, Ann. I. H. Poincaré, 21 (2004), 381-399. doi: 10.1016/j.anihpc.2003.05.002. Google Scholar [17] H. Ohtsuka, T. Ricciardi and T. Suzuki, Blow-up analysis for an elliptic equation describing stationary vortex flows with variable intensities in 2D-turbulence, J. Differential Equations, 249 (2010), 1436-1465. doi: 10.1016/j.jde.2010.06.006. Google Scholar [18] H. Ohtsuka and T. Suzuki, Mean field equation for the equilibrium turbulence and a related functional inequality, Adv. Differential Equations, 11 (2006), 281-304. Google Scholar [19] L. Onsager, Statistical hydrodynamics, Nuovo Cimento Suppl, 6 (1949), 279-287. doi: 10.1007/BF02780991. Google Scholar [20] A. Pistoia and T. Ricciardi, Concentrating solutions for a Liouville type equation with variable intensities in 2D-turbulence, Nonlinearity, 29 (2016), 271-297. doi: 10.1088/0951-7715/29/2/271. Google Scholar [21] Y. B. Pointin and T. S. Lundgren, Statistical mechanics of two-dimensional vortices in a bounded container, Phys. Fluids, 19 (1976), 1459-1470. Google Scholar [22] J. Prajapat and G. Tarantello, On a class of elliptic problems in $\mathbb R^2$: Symmetry and uniqueness results, Proc. R. Soc. Edinb. Sect. A, 131 (2001), 967-985. doi: 10.1017/S0308210500001219. Google Scholar [23] T. Ricciardi, Mountain-pass solutions for a mean field equation from two-dimensional turbulence, Differential and Integral Equations, 20 (2007), 561-575. Google Scholar [24] T. Ricciardi and G. Zecca, Minimal blow-up masses and existence of solutions for an asymmetric sinh-Poisson equation, arXiv: 1605.05895Google Scholar [25] 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. Google Scholar [26] R. Takahashi, Analysis Seminar, Naples Federico Ⅱ University, March 2016.Google Scholar [27] N. S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-483. Google Scholar [28] D. Ye, Une remarque sur le comportement asymptotique des solutions de $-Δ u=λ f(u)$, C.R. Acad. Sci. Paris, 325 (1997), 1279-1282. doi: 10.1016/S0764-4442(97)82353-1. Google Scholar [29] C. Zhou, Existence of solution for mean field equation for the equilibrium turbulence, Nonlinear Anal., 69 (2008), 2541-2552. doi: 10.1016/j.na.2007.08.029. Google Scholar

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##### References:
 [1] 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, Commun. Math. Phys., 174 (1995), 229-260. doi: 10.1007/BF02099602. Google Scholar [2] M. del Pino, P. Esposito and M. Musso, Nondegeneracy of entire solutions of a singular Liouville equation, Proc. Am. Math. Soc., 140 (2012), 581-588. doi: 10.1090/S0002-9939-2011-11134-1. Google Scholar [3] M. del Pino, M. Kowalczyk and M. Musso, Singular limits in Liouville-type equations, Calc. Var. Partial Differential Equations, 24 (2005), 47-81. doi: 10.1007/s00526-004-0314-5. Google Scholar [4] P. Esposito, M. Grossi and A. Pistoia, On the existence of blowing-up solutions for a mean field equation, Ann. I. H. Poincaré, 22 (2005), 227-257. doi: 10.1016/j.anihpc.2004.12.001. Google Scholar [5] G. L. Eyink and K. R. Sreenivasan, Onsager and the theory of hydrodynamic turbulence, Reviews of Modern Physics, 78 (2006), 87-135. doi: 10.1103/RevModPhys.78.87. Google Scholar [6] M. Grossi, C. Grumiau and F. Pacella, Lane Emden problems with large exponents and singular Liouville equations, J. Math. Pures Appl., 101 (2014), 735-754. doi: 10.1016/j.matpur.2013.06.011. Google Scholar [7] M. Grossi and A. Pistoia, Multiple blow-up phenomena for the sinh-Poisson equation, Arch. Rational Mech. Anal., 209 (2013), 287-320. doi: 10.1007/s00205-013-0625-9. Google Scholar [8] A. Jevnikar, An existence result for the mean-field equation on compact surfaces in a doubly supercritical regime, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 1021-1045. doi: 10.1017/S030821051200042X. Google Scholar [9] A. Jevnikar and W. Yang, Analytic aspects of the Tzitzéica equation: Blow-up analysis and existence results, Calc. Var. Partial Differential Equations, 56 (2017), Paper No. 43, 38 pp. doi: 10.1007/s00526-017-1136-6. Google Scholar [10] D. D. Joseph and T. S. Lundgren, Quasilinear problems driven by positive sources, Arch. Rat. Mech. Anal., 49 (1973), 241-269. doi: 10.1007/BF00250508. Google Scholar [11] J. Jost, G. Wang, D. Ye and C. Zhou, The blow up analysis of solutions of the elliptic sinh-Gordon equation, Calc. Var. Partial Differential Equations, 31 (2008), 263-276. doi: 10.1007/s00526-007-0116-7. Google Scholar [12] G. Joyce and D. Montgomery, Negative temperature states for the two-dimensional guiding centre plasma, J. Plasma Phys., 10 (1973), 107-121. Google Scholar [13] C. S. Lin, An expository survey on recent development of mean field equations, Discr. Cont. Dynamical Systems, 19 (2007), 387-410. doi: 10.3934/dcds.2007.19.387. Google Scholar [14] A. Malchiodi, Topological methods for an elliptic equation with exponential nonlinearities, Discr. Cont. Dynamical Systems, 21 (2008), 277-294. doi: 10.3934/dcds.2008.21.277. Google Scholar [15] J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J, 20 (1970/71), 1077-1092. doi: 10.1512/iumj.1971.20.20101. Google Scholar [16] C. Neri, Statistical Mechanics of the $N$-point vortex system with random intesities on a bounded domain, Ann. I. H. Poincaré, 21 (2004), 381-399. doi: 10.1016/j.anihpc.2003.05.002. Google Scholar [17] H. Ohtsuka, T. Ricciardi and T. Suzuki, Blow-up analysis for an elliptic equation describing stationary vortex flows with variable intensities in 2D-turbulence, J. Differential Equations, 249 (2010), 1436-1465. doi: 10.1016/j.jde.2010.06.006. Google Scholar [18] H. Ohtsuka and T. Suzuki, Mean field equation for the equilibrium turbulence and a related functional inequality, Adv. Differential Equations, 11 (2006), 281-304. Google Scholar [19] L. Onsager, Statistical hydrodynamics, Nuovo Cimento Suppl, 6 (1949), 279-287. doi: 10.1007/BF02780991. Google Scholar [20] A. Pistoia and T. Ricciardi, Concentrating solutions for a Liouville type equation with variable intensities in 2D-turbulence, Nonlinearity, 29 (2016), 271-297. doi: 10.1088/0951-7715/29/2/271. Google Scholar [21] Y. B. Pointin and T. S. Lundgren, Statistical mechanics of two-dimensional vortices in a bounded container, Phys. Fluids, 19 (1976), 1459-1470. Google Scholar [22] J. Prajapat and G. Tarantello, On a class of elliptic problems in $\mathbb R^2$: Symmetry and uniqueness results, Proc. R. Soc. Edinb. Sect. A, 131 (2001), 967-985. doi: 10.1017/S0308210500001219. Google Scholar [23] T. Ricciardi, Mountain-pass solutions for a mean field equation from two-dimensional turbulence, Differential and Integral Equations, 20 (2007), 561-575. Google Scholar [24] T. Ricciardi and G. Zecca, Minimal blow-up masses and existence of solutions for an asymmetric sinh-Poisson equation, arXiv: 1605.05895Google Scholar [25] 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. Google Scholar [26] R. Takahashi, Analysis Seminar, Naples Federico Ⅱ University, March 2016.Google Scholar [27] N. S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-483. Google Scholar [28] D. Ye, Une remarque sur le comportement asymptotique des solutions de $-Δ u=λ f(u)$, C.R. Acad. Sci. Paris, 325 (1997), 1279-1282. doi: 10.1016/S0764-4442(97)82353-1. Google Scholar [29] C. Zhou, Existence of solution for mean field equation for the equilibrium turbulence, Nonlinear Anal., 69 (2008), 2541-2552. doi: 10.1016/j.na.2007.08.029. Google Scholar
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