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Classification of radial solutions to Liouville systems with singularities
1. | Taida Institute of Mathematical Sciences and Center for Advanced Study in Theoretical Sciences, National Taiwan University, Taipei 106, Taiwan |
2. | Department of Mathematics, University of Florida, 358 Little Hall, P.O.Box 118105, Gainesville, Florida 32611-8105, United States |
References:
[1] |
D. Bartolucci, C.-C. Chen, C.-S. Lin and G. Tarantello, Profile of blow-up solutions to mean field equations with singular data, Comm. Partial Differential Equations, 29 (2004), 1241-1265.
doi: 10.1081/PDE-200033739. |
[2] |
W. H. Bennet, Magnetically self-focusing streams, Phys. Rev., 45 (1934), 890-897. |
[3] |
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. |
[4] |
S.-Y. Chang, M. Gursky and P. Yang, The scalar curvature equation on 2- and 3-spheres, Calc. Var. and PDE, 1 (1993), 205-229.
doi: 10.1007/BF01191617. |
[5] |
S.-Y. Chang and P. Yang, Prescribing Gaussian curvatuare on S2, Acta Math., 159 (1987), 215-259.
doi: 10.1007/BF02392560. |
[6] |
C.-C. Chen and C.-S. Lin, Estimate of the conformal scalar curvature equation via the method of moving planes. II, J. Differential Geom., 49 (1998), 115-178. |
[7] |
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.
doi: 10.1002/cpa.3014. |
[8] |
C.-C. Chen and C.-S. Lin, Mean field equations of Liouville type with singular data: Sharper estimates, Discrete and Continuous Dynamic Systems, 28 (2010), 1237-1272.
doi: 10.3934/dcds.2010.28.1237. |
[9] |
W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622.
doi: 10.1215/S0012-7094-91-06325-8. |
[10] |
W. X. Chen and C. Li, Qualitative properties of solutions to some nonlinear elliptic equations in $\mathbb R^2$, Duke Math. J., 71 (1993), 427-439.
doi: 10.1215/S0012-7094-93-07117-7. |
[11] |
S. Childress and J. K. Percus, Nonlinear aspects of chemotaxis, Math. Biosci., 56 (1981), 217-237.
doi: 10.1016/0025-5564(81)90055-9. |
[12] |
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. |
[13] |
M. Chipot, I. Shafrir and G. Wolansky, Erratum: "On the solutions of Liouville systems'' [J. Differential Equations, 140 (1997), 59-105; MR1473855], J. Differential Equations, 178 (2002), 630. |
[14] |
P. Debye and E. Huckel, Zur theorie der electrolyte, Phys. Zft, 24 (1923), 305-325. |
[15] |
J. Hong, Y. Kim and P. Y. Pac, Multivortex solutions of the abelian Chern-Simons-Higgs theory, Phys. Rev. Letter, 64 (1990), 2230-2233.
doi: 10.1103/PhysRevLett.64.2230. |
[16] |
R. Jackiw and E. J. Weinberg, Selfdual Chern Simons vortices, Phys. Rev. Lett., 64 (1990), 2234-2237.
doi: 10.1103/PhysRevLett.64.2234. |
[17] |
J. Jost, C. 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. |
[18] |
J. Jost and G. Wang, Classification of solutions of a Toda system in $\mathbb R^ 2$,, Int. Math. Res. Not., 2002 (): 277.
doi: 10.1155/S1073792802105022. |
[19] |
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. |
[20] |
J. Kazdan and F. Warner, Curvature functions for compact 2-manifolds, Ann. of Math. (2), 99 (1974), 14-47.
doi: 10.2307/1971012. |
[21] |
E. F. Keller and L. A. Segel, Traveling bands of Chemotactic Bacteria: A theoretical analysis, J. Theor. Biol., 30 (1971), 235-248.
doi: 10.1016/0022-5193(71)90051-8. |
[22] |
M. K.-H. Kiessling and J. L. Lebowitz, Dissipative stationary plasmas: Kinetic modeling, Bennett's pinch and generalizations, Phys. Plasmas, 1 (1994), 1841-1849.
doi: 10.1063/1.870639. |
[23] |
Y. Y. Li, Harnack type inequality: The method of moving planes, Comm. Math. Phys., 200 (1999), 421-444.
doi: 10.1007/s002200050536. |
[24] |
C.-S. Lin and L. Zhang, Profile of bubbling solutions to a Liouville system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 117-143.
doi: 10.1016/j.anihpc.2009.09.001. |
[25] |
C.-S. Lin and L. Zhang, A topological degree counting for some Liouville systems of mean field type, Comm. Pure Appl. Math., 64 (2011), 556-590.
doi: 10.1002/cpa.20355. |
[26] |
M. S. Mock, Asymptotic behavior of solutions of transport equations for semiconductor devices, J. Math. Anal. Appl., 49 (1975), 215-225.
doi: 10.1016/0022-247X(75)90172-9. |
[27] |
M. Nolasco and G. Tarantello, Double vortex condensates in the Chern-Simons-Higgs theory, Calc. Var. and PDE, 9 (1999), 31-34.
doi: 10.1007/s005260050132. |
[28] |
J. Prajapat and G. Tarantello, On a class of elliptic problems in R2: Symmetry and uniqueness results, Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), 967-985.
doi: 10.1017/S0308210500001219. |
[29] |
I. Rubinstein, Electro Diffusion of Ions, SIAM Studies in Applied Mathematics, 11, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1990.
doi: 10.1137/1.9781611970814. |
[30] |
L. Zhang, Blow up solutions of some nonlinear elliptic equations involving exponential nonlinearities, Comm. Math. Phys., 268 (2006), 105-133.
doi: 10.1007/s00220-006-0092-3. |
[31] |
L. Zhang, Asymptotic behavior of blowup solutions for elliptic equations with exponential nonlinearity and singular data, Commun. Contemp. Math., 11 (2009), 395-411.
doi: 10.1142/S0219199709003417. |
show all references
References:
[1] |
D. Bartolucci, C.-C. Chen, C.-S. Lin and G. Tarantello, Profile of blow-up solutions to mean field equations with singular data, Comm. Partial Differential Equations, 29 (2004), 1241-1265.
doi: 10.1081/PDE-200033739. |
[2] |
W. H. Bennet, Magnetically self-focusing streams, Phys. Rev., 45 (1934), 890-897. |
[3] |
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. |
[4] |
S.-Y. Chang, M. Gursky and P. Yang, The scalar curvature equation on 2- and 3-spheres, Calc. Var. and PDE, 1 (1993), 205-229.
doi: 10.1007/BF01191617. |
[5] |
S.-Y. Chang and P. Yang, Prescribing Gaussian curvatuare on S2, Acta Math., 159 (1987), 215-259.
doi: 10.1007/BF02392560. |
[6] |
C.-C. Chen and C.-S. Lin, Estimate of the conformal scalar curvature equation via the method of moving planes. II, J. Differential Geom., 49 (1998), 115-178. |
[7] |
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.
doi: 10.1002/cpa.3014. |
[8] |
C.-C. Chen and C.-S. Lin, Mean field equations of Liouville type with singular data: Sharper estimates, Discrete and Continuous Dynamic Systems, 28 (2010), 1237-1272.
doi: 10.3934/dcds.2010.28.1237. |
[9] |
W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622.
doi: 10.1215/S0012-7094-91-06325-8. |
[10] |
W. X. Chen and C. Li, Qualitative properties of solutions to some nonlinear elliptic equations in $\mathbb R^2$, Duke Math. J., 71 (1993), 427-439.
doi: 10.1215/S0012-7094-93-07117-7. |
[11] |
S. Childress and J. K. Percus, Nonlinear aspects of chemotaxis, Math. Biosci., 56 (1981), 217-237.
doi: 10.1016/0025-5564(81)90055-9. |
[12] |
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. |
[13] |
M. Chipot, I. Shafrir and G. Wolansky, Erratum: "On the solutions of Liouville systems'' [J. Differential Equations, 140 (1997), 59-105; MR1473855], J. Differential Equations, 178 (2002), 630. |
[14] |
P. Debye and E. Huckel, Zur theorie der electrolyte, Phys. Zft, 24 (1923), 305-325. |
[15] |
J. Hong, Y. Kim and P. Y. Pac, Multivortex solutions of the abelian Chern-Simons-Higgs theory, Phys. Rev. Letter, 64 (1990), 2230-2233.
doi: 10.1103/PhysRevLett.64.2230. |
[16] |
R. Jackiw and E. J. Weinberg, Selfdual Chern Simons vortices, Phys. Rev. Lett., 64 (1990), 2234-2237.
doi: 10.1103/PhysRevLett.64.2234. |
[17] |
J. Jost, C. 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. |
[18] |
J. Jost and G. Wang, Classification of solutions of a Toda system in $\mathbb R^ 2$,, Int. Math. Res. Not., 2002 (): 277.
doi: 10.1155/S1073792802105022. |
[19] |
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. |
[20] |
J. Kazdan and F. Warner, Curvature functions for compact 2-manifolds, Ann. of Math. (2), 99 (1974), 14-47.
doi: 10.2307/1971012. |
[21] |
E. F. Keller and L. A. Segel, Traveling bands of Chemotactic Bacteria: A theoretical analysis, J. Theor. Biol., 30 (1971), 235-248.
doi: 10.1016/0022-5193(71)90051-8. |
[22] |
M. K.-H. Kiessling and J. L. Lebowitz, Dissipative stationary plasmas: Kinetic modeling, Bennett's pinch and generalizations, Phys. Plasmas, 1 (1994), 1841-1849.
doi: 10.1063/1.870639. |
[23] |
Y. Y. Li, Harnack type inequality: The method of moving planes, Comm. Math. Phys., 200 (1999), 421-444.
doi: 10.1007/s002200050536. |
[24] |
C.-S. Lin and L. Zhang, Profile of bubbling solutions to a Liouville system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 117-143.
doi: 10.1016/j.anihpc.2009.09.001. |
[25] |
C.-S. Lin and L. Zhang, A topological degree counting for some Liouville systems of mean field type, Comm. Pure Appl. Math., 64 (2011), 556-590.
doi: 10.1002/cpa.20355. |
[26] |
M. S. Mock, Asymptotic behavior of solutions of transport equations for semiconductor devices, J. Math. Anal. Appl., 49 (1975), 215-225.
doi: 10.1016/0022-247X(75)90172-9. |
[27] |
M. Nolasco and G. Tarantello, Double vortex condensates in the Chern-Simons-Higgs theory, Calc. Var. and PDE, 9 (1999), 31-34.
doi: 10.1007/s005260050132. |
[28] |
J. Prajapat and G. Tarantello, On a class of elliptic problems in R2: Symmetry and uniqueness results, Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), 967-985.
doi: 10.1017/S0308210500001219. |
[29] |
I. Rubinstein, Electro Diffusion of Ions, SIAM Studies in Applied Mathematics, 11, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1990.
doi: 10.1137/1.9781611970814. |
[30] |
L. Zhang, Blow up solutions of some nonlinear elliptic equations involving exponential nonlinearities, Comm. Math. Phys., 268 (2006), 105-133.
doi: 10.1007/s00220-006-0092-3. |
[31] |
L. Zhang, Asymptotic behavior of blowup solutions for elliptic equations with exponential nonlinearity and singular data, Commun. Contemp. Math., 11 (2009), 395-411.
doi: 10.1142/S0219199709003417. |
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