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Morse indices and the number of blow up points of blowing-up solutions for a Liouville equation with singular data
1. | Department of Mathematics, Osaka City University, 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka, 558-8585 |
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 no. 7-8 (2004), 29 (2004), 7.
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[2] |
D. Bartolucci, and G. Tarantello:, The Liouville equation with singular data: a concentration-compactness principle via a local representation formula,, J. Differential Equations 185 (2002), 185 (2002), 161.
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[3] |
D. Bartolucci, and G. Tarantello:, Liouville type equations with singular data and their applications to periodic multivortices for the Electroweak Theory,, Comm. Math. Pfys. 229 (2002), 229 (2002), 3.
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[4] |
H. Brezis, and F. Merle:, Uniform estimates and blow-up behavior for solutions of $-\Delta u = V(x)e^u$ in two dimensions,, Comm. Partial Differential Equations 16 (1991), 16 (1991), 1223.
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[5] |
P. Esposito:, A Class of Liouville-Type Equations Arising in Chern-Simons Vortex Theory: Asymptotics and Construction of Blowing Up Solutions,, Ph. D. thesis, (2003). Google Scholar |
[6] |
P. Esposito:, Blowup solutions for a Liouville equation with singular data,, SIAM. J. Math. Anal. 36 (2005), 36 (2005), 1310.
|
[7] |
P. Esposito:, Blowup solutions for a Liouville equation with singular data,, in Proceedings of the International Conference, (2005), 61.
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[8] |
Y. Y. Li, and I. Shafrir:, Blow-up analysis for solutions of $-\Delta u = V e^u$ in dimension two,, Indiana Univ. Math. J. 43 (1994), 43 (1994), 1255.
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[9] |
L. Ma, and J. Wei:, Convergence for a Liouville equation,, Comment. Math. Helv. 76 (2001), 76 (2001), 506.
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[10] |
K. Nagasaki, and T. Suzuki:, Asymptotic analysis for two-dimensional elliptic eigenvalue problems with exponentially dominated nonlinearities,, Asymptotic Anal. 3 (1990), 3 (1990), 173.
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[11] |
J. Prajapat, and G. Tarantello:, On a class of elliptic problems in $\mathbbR^2$: symmetry and uniqueness results,, Proc. Roy. Soc. Edinburgh 131 A (2001), 131 A (2001), 967.
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[12] |
F. Takahashi:, Blow up points and the Morse indices of solutions to the Liouville equation in two-dimension,, Advances in Nonlinear Stud. 12 no.1, 12 (2012), 115.
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[13] |
F. Takahashi:, Blow up points and the Morse indices of solutions to the Liouville equation : inhomogeneous case,, submitted., (). Google Scholar |
[14] |
G. Tarantello:, " Selfdual Gauge Field Vortices: An Analytical Approach,", Progress in Nonlinear Differential Equations and Their Applications 72, (2008).
|
[15] |
Y. Yang:, "Solitons in Field Theory and Nonlinear Analysis,", Springer Monographs in Mathematics, (2001).
|
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 no. 7-8 (2004), 29 (2004), 7.
|
[2] |
D. Bartolucci, and G. Tarantello:, The Liouville equation with singular data: a concentration-compactness principle via a local representation formula,, J. Differential Equations 185 (2002), 185 (2002), 161.
|
[3] |
D. Bartolucci, and G. Tarantello:, Liouville type equations with singular data and their applications to periodic multivortices for the Electroweak Theory,, Comm. Math. Pfys. 229 (2002), 229 (2002), 3.
|
[4] |
H. Brezis, and F. Merle:, Uniform estimates and blow-up behavior for solutions of $-\Delta u = V(x)e^u$ in two dimensions,, Comm. Partial Differential Equations 16 (1991), 16 (1991), 1223.
|
[5] |
P. Esposito:, A Class of Liouville-Type Equations Arising in Chern-Simons Vortex Theory: Asymptotics and Construction of Blowing Up Solutions,, Ph. D. thesis, (2003). Google Scholar |
[6] |
P. Esposito:, Blowup solutions for a Liouville equation with singular data,, SIAM. J. Math. Anal. 36 (2005), 36 (2005), 1310.
|
[7] |
P. Esposito:, Blowup solutions for a Liouville equation with singular data,, in Proceedings of the International Conference, (2005), 61.
|
[8] |
Y. Y. Li, and I. Shafrir:, Blow-up analysis for solutions of $-\Delta u = V e^u$ in dimension two,, Indiana Univ. Math. J. 43 (1994), 43 (1994), 1255.
|
[9] |
L. Ma, and J. Wei:, Convergence for a Liouville equation,, Comment. Math. Helv. 76 (2001), 76 (2001), 506.
|
[10] |
K. Nagasaki, and T. Suzuki:, Asymptotic analysis for two-dimensional elliptic eigenvalue problems with exponentially dominated nonlinearities,, Asymptotic Anal. 3 (1990), 3 (1990), 173.
|
[11] |
J. Prajapat, and G. Tarantello:, On a class of elliptic problems in $\mathbbR^2$: symmetry and uniqueness results,, Proc. Roy. Soc. Edinburgh 131 A (2001), 131 A (2001), 967.
|
[12] |
F. Takahashi:, Blow up points and the Morse indices of solutions to the Liouville equation in two-dimension,, Advances in Nonlinear Stud. 12 no.1, 12 (2012), 115.
|
[13] |
F. Takahashi:, Blow up points and the Morse indices of solutions to the Liouville equation : inhomogeneous case,, submitted., (). Google Scholar |
[14] |
G. Tarantello:, " Selfdual Gauge Field Vortices: An Analytical Approach,", Progress in Nonlinear Differential Equations and Their Applications 72, (2008).
|
[15] |
Y. Yang:, "Solitons in Field Theory and Nonlinear Analysis,", Springer Monographs in Mathematics, (2001).
|
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