# American Institute of Mathematical Sciences

June  2013, 33(6): 2299-2318. doi: 10.3934/dcds.2013.33.2299

## Structure of solutions to a singular Liouville system arising from modeling dissipative stationary plasmas

 1 Department of Mathematics, National Central University, Chung-Li 32001, Taiwan, Taiwan 2 Mathematics Division, National Center for Theoretical Sciences, Hsinchu 30013, Taiwan

Received  October 2011 Revised  October 2012 Published  December 2012

Arising from one-particle distribution functions of stationary dissipative plasmas, we consider a coupled elliptic system with singular data in the plane. The existence and uniqueness of solutions to the Dirichlet boundary value problem are proved. In addition, the structure of other solutions, including blow-up solutions, is also clarified.
Citation: Jann-Long Chern, Zhi-You Chen, Yong-Li Tang. Structure of solutions to a singular Liouville system arising from modeling dissipative stationary plasmas. Discrete and Continuous Dynamical Systems, 2013, 33 (6) : 2299-2318. doi: 10.3934/dcds.2013.33.2299
##### References:
 [1] W. H. Bennet, Magnetically self-focusing streams, Phys. Rev., 45 (1934), 890-897. [2] J.-L. Chern, Z.-Y. Chen and C.-S. Lin, Uniqueness of topological solutions and the structure of solutions for the Chern-Simons system with two Higgs particles, Comm. Math. Phys., 296 (2010), 323-351. doi: 10.1007/s00220-010-1021-z. [3] Z.-Y. Chen, J.-L. Chern and Y.-L. Tang, On the solutions to a Liouville-type system involving singularity, Calc. Var. Partial Differential Equations, 43 (2012), 57-81. doi: 10.1007/s00526-011-0403-1. [4] Z.-Y. Chen, J.-L. Chern, J. Shi and Y.-L. Tang, On the uniqueness and structure of solutions to a coupled elliptic system, J. Differential Equations, 249 (2010), 3419-3442. doi: 10.1016/j.jde.2010.09.001. [5] J.-L. Chern, Z.-Y. Chen, Y.-L. Tang and C.-S. Lin, Uniqueness and structure of solutions to the Dirichlet problem for an elliptic system, J. Differential Equations, 246 (2009), 3704-3714. doi: 10.1016/j.jde.2009.01.005. [6] S. Chanillo and M. K.-H. Kiessling, Conformally invariant systems of nonlinear PDE of Liouville type, Geom. Funct. Anal., 5 (1995), 924-947. doi: 10.1007/BF01902215. [7] S. Childress and J. K. Percus, Nonlinear aspects of Chemotaxis, Math. Biosci., 56 (1981), 217-237. doi: 10.1016/0025-5564(81)90055-9. [8] M. Chipot, I. Shafrir and G. Wolansky, On the solutions of Liouville systems, J. Differential Equations, 140 (1997), 59-105. doi: 10.1006/jdeq.1997.3316. [9] G. Dunne, "Self-dual Chern-Simons Theories," Lecture Notes in Physics, m36, Berlin: Springer-Verlag, 1995. [10] P. Debye and E. Huckel, Zur theorie der electrolyte, Phys. Zft, 24 (1923), 305-325. [11] J. Jost, C. S. Lin and G. Wang, Analytic aspects of the Toda system. II. Bubbling behavior and existence of solutions, Comm. Pure Appl. Math., 59 (2006), 526-558. doi: 10.1002/cpa.20099. [12] J. Jost and G. Wang, Classification of solutions of a Toda system in $\mathbb{R}^2$, Int. Math. Res. Not., (2002), 277-290. doi: 10.1155/S1073792802105022. [13] J. Jost and G. Wang, Analytic aspects of the Toda system. I. A Moser-Trudinger inequality, Comm. Pure Appl. Math., 54 (2001), 1289-1319. doi: 10.1002/cpa.10004. [14] M. K.-H. Kiessling, Symmetry results for finite-temperature, relativistic Thomas-Fermi equations, Comm. Math. Phys., 226 (2002), 607-626. doi: 10.1007/s002200200625. [15] M. K.-H. Kiessling and J. L. Lebowitz, Dissipative stationary Plasmas: Kinetic Modeling Bennet Pinch, and generalizations, Phys. Plasmas, 1 (1994), 1841-1849. doi: 10.1063/1.870639. [16] E. F. Keller and L. A. Segel, Traveling bands of Chemotactic Bacteria: A theoretical analysis, J. Theor. Biol., 30 (1971), 235-248. [17] C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Invent. Math., 123 (1996), 221-231. doi: 10.1007/s002220050023. [18] C.-S. Lin and L. Zhang, Profile of bubbling solutions to a Liouville system, Ann. I. H. Poincaré-AN, 27 (2010), 117-143. doi: 10.1016/j.anihpc.2009.09.001. [19] M. S. Mock, Asymptotic behavior of solutions of transport equations for semiconductor devices, J. Math. Anal. Appl., 49 (1975), 215-225. [20] Y. Yang, "Solitons in Field Theory and Nonlinear Analysis," Springer-Verlag, 2001.

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##### References:
 [1] W. H. Bennet, Magnetically self-focusing streams, Phys. Rev., 45 (1934), 890-897. [2] J.-L. Chern, Z.-Y. Chen and C.-S. Lin, Uniqueness of topological solutions and the structure of solutions for the Chern-Simons system with two Higgs particles, Comm. Math. Phys., 296 (2010), 323-351. doi: 10.1007/s00220-010-1021-z. [3] Z.-Y. Chen, J.-L. Chern and Y.-L. Tang, On the solutions to a Liouville-type system involving singularity, Calc. Var. Partial Differential Equations, 43 (2012), 57-81. doi: 10.1007/s00526-011-0403-1. [4] Z.-Y. Chen, J.-L. Chern, J. Shi and Y.-L. Tang, On the uniqueness and structure of solutions to a coupled elliptic system, J. Differential Equations, 249 (2010), 3419-3442. doi: 10.1016/j.jde.2010.09.001. [5] J.-L. Chern, Z.-Y. Chen, Y.-L. Tang and C.-S. Lin, Uniqueness and structure of solutions to the Dirichlet problem for an elliptic system, J. Differential Equations, 246 (2009), 3704-3714. doi: 10.1016/j.jde.2009.01.005. [6] S. Chanillo and M. K.-H. Kiessling, Conformally invariant systems of nonlinear PDE of Liouville type, Geom. Funct. Anal., 5 (1995), 924-947. doi: 10.1007/BF01902215. [7] S. Childress and J. K. Percus, Nonlinear aspects of Chemotaxis, Math. Biosci., 56 (1981), 217-237. doi: 10.1016/0025-5564(81)90055-9. [8] M. Chipot, I. Shafrir and G. Wolansky, On the solutions of Liouville systems, J. Differential Equations, 140 (1997), 59-105. doi: 10.1006/jdeq.1997.3316. [9] G. Dunne, "Self-dual Chern-Simons Theories," Lecture Notes in Physics, m36, Berlin: Springer-Verlag, 1995. [10] P. Debye and E. Huckel, Zur theorie der electrolyte, Phys. Zft, 24 (1923), 305-325. [11] J. Jost, C. S. Lin and G. Wang, Analytic aspects of the Toda system. II. Bubbling behavior and existence of solutions, Comm. Pure Appl. Math., 59 (2006), 526-558. doi: 10.1002/cpa.20099. [12] J. Jost and G. Wang, Classification of solutions of a Toda system in $\mathbb{R}^2$, Int. Math. Res. Not., (2002), 277-290. doi: 10.1155/S1073792802105022. [13] J. Jost and G. Wang, Analytic aspects of the Toda system. I. A Moser-Trudinger inequality, Comm. Pure Appl. Math., 54 (2001), 1289-1319. doi: 10.1002/cpa.10004. [14] M. K.-H. Kiessling, Symmetry results for finite-temperature, relativistic Thomas-Fermi equations, Comm. Math. Phys., 226 (2002), 607-626. doi: 10.1007/s002200200625. [15] M. K.-H. Kiessling and J. L. Lebowitz, Dissipative stationary Plasmas: Kinetic Modeling Bennet Pinch, and generalizations, Phys. Plasmas, 1 (1994), 1841-1849. doi: 10.1063/1.870639. [16] E. F. Keller and L. A. Segel, Traveling bands of Chemotactic Bacteria: A theoretical analysis, J. Theor. Biol., 30 (1971), 235-248. [17] C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Invent. Math., 123 (1996), 221-231. doi: 10.1007/s002220050023. [18] C.-S. Lin and L. Zhang, Profile of bubbling solutions to a Liouville system, Ann. I. H. Poincaré-AN, 27 (2010), 117-143. doi: 10.1016/j.anihpc.2009.09.001. [19] M. S. Mock, Asymptotic behavior of solutions of transport equations for semiconductor devices, J. Math. Anal. Appl., 49 (1975), 215-225. [20] Y. Yang, "Solitons in Field Theory and Nonlinear Analysis," Springer-Verlag, 2001.
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