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The Maslov and Morse indices for Sturm-Liouville systems on the half-line
Singular limit solutions for a 2-dimensional semilinear elliptic system of Liouville type in some general case
1. | University of Tunis El Manar, Faculty of Sciences of Tunis, Department of Mathematics, Campus University 2092 Tunis, Tunisia |
2. | University of Tunis El Manar, higher Institute of medical technologies of Tunis, 9 street Dr. Zouhair Essafi 1006 Tunis, Tunisia |
The existence of singular limit solutions are investigated by establishing a new Liouville type theorem for nonlinear elliptic system by using the Pohozaev type identity and the nonlinear domain decomposition method.
References:
[1] |
N. Akhmediev and A. Ankiewicz,
Partially coherent solitons on a finite background, Phys.Rev. Lett, 82 (1999), 2661-2664.
doi: 10.1103/PhysRevLett.82.2661. |
[2] |
S. Baraket, I. Ben Omrane, T. Ouni and N. Trabelsi,
Singular limits for 2-dimensional elliptic problem with exponentially dominated nonlinearity and singular data, Commun. Contemp. Math, 13 (2011), 697-725.
doi: 10.1142/S0219199711004282. |
[3] |
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.
doi: 10.1007/s005260050080. |
[4] |
S. Baraket and D. Ye,
Singular limit solutions for two-dimentional elliptic problems with exponentionally dominated nonlinearity, Chinese Ann. Math. Ser. B, 22 (2001), 287-296.
doi: 10.1142/S0252959901000309. |
[5] |
S. Baraket, M. Dammak, T. Ouni and F. Pacard,
Singular limits for a 4-dimensional semilinear elliptic problem with exponential nonlinearity, Ann. I. H. Poincaré - AN, 24 (2007), 875-895.
doi: 10.1016/j.anihpc.2006.06.009. |
[6] |
D. Bartolucci and G. Tarantello,
Liouville type equations with singular data and their applications to periodic multivortices for the Electroweak Theory, Comm. Math. Phys, 229 (2002), 3-47.
doi: 10.1007/s002200200664. |
[7] |
W. H. Bonnett,
Magntically self-focussing streams, Phys. Rev, 45 (1934), 890-897.
|
[8] |
S. Chanillo and M. K. H. Kiessling,
Conformaly invariant systems of nonlinear PDE of Liouville type, Geom. Funct. Anal, 5 (1995), 924-947.
doi: 10.1007/BF01902215. |
[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] |
Z. Chen and W. Zou,
On coupled systems of Schrödinger equations, Adv.Differ. Equ, 16 (2011), 775-800.
|
[11] |
Z. Chen and W. Zou,
An optimal constant for the existence of least energy solutions of a coupled Schrödinger system, Calc. Var. Partial Differ. Equ, 48 (2013), 695-711.
doi: 10.1007/s00526-012-0568-2. |
[12] |
Z. Chen and W. Zou,
On linearly coupled Schrödinger systems, Proc.Am. Math. Soc, 142 (2014), 323-333.
doi: 10.1090/S0002-9939-2013-12000-9. |
[13] |
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. |
[14] |
P. Esposito,
Blow up solutions for a Liouville equation with singular data, SIAM J. Math. Anal, 36 (2005), 1310-1345.
doi: 10.1137/S0036141003430548. |
[15] |
B. D. Esry, C. H. Greene, J. P. Burke and J. L. Bohn Jr.,
Hartree-Fock theory for double condensates, Phys. Rev. Lett, 78 (1997), 3594-3597.
doi: 10.1103/PhysRevLett.78.3594. |
[16] |
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. |
[17] |
C.-S. Lin and G. Tarantello,
When blow-up does not imply concentration: A detour from Brézis-Merle's result, Comptes Rendus Mathematique, 354 (2016), 493-498.
doi: 10.1016/j.crma.2016.01.014. |
[18] |
J. Liouville,
Sur l'équation aux différences partielles $\partial^{2}\log\frac{\lambda}{\partial u \partial v}\pm\frac{\lambda}{2a^{2}} = 0 $, J. Math, 18 (1853), 17-72.
|
[19] |
J. Liu, Y. Guo and Y. Zhang,
Liouville-type theorems for polyharmonic systems in RN, J. Differential Equations, 225 (2006), 685-709.
doi: 10.1016/j.jde.2005.10.016. |
[20] |
C.-S. Lin, J. C. Wei and D. Ye,
Classifcation and nondegeneracy of SU(n+1) Toda system, Invent. Math, 190 (2012), 169-207.
doi: 10.1007/s00222-012-0378-3. |
[21] |
C.-S. Lin, J. Wei and C. Zhao,
Classification of blow-up limits for SU(3) singular Toda systems, Analysis and PDE, 8 (2015), 807-837.
doi: 10.2140/apde.2015.8.807. |
[22] |
M. Musso, A. Pistoia and J. Wei,
New blow-up phenomena for SU(n+1) Toda system, J. Differential Equations, 260 (2016), 6232-6266.
doi: 10.1016/j.jde.2015.12.036. |
[23] |
N. Nagasaki and T. Suzuki,
Asymptotic analysis for a two dimensional elliptic eigenvalue problem with dominated nonlinearity, Asymptotic Analysis, 3 (1990), 173-188.
|
[24] |
T. Suzuki, Two dimensional Emden-Fowler equation with exponential nonlinearity, Nonlinear Diffusion Equations and Their Equilibrium Statesd, 3, (Gregynog, 1989), Birkäuser Boston, Boston, MA, 7 (1992), 493–512. |
show all references
References:
[1] |
N. Akhmediev and A. Ankiewicz,
Partially coherent solitons on a finite background, Phys.Rev. Lett, 82 (1999), 2661-2664.
doi: 10.1103/PhysRevLett.82.2661. |
[2] |
S. Baraket, I. Ben Omrane, T. Ouni and N. Trabelsi,
Singular limits for 2-dimensional elliptic problem with exponentially dominated nonlinearity and singular data, Commun. Contemp. Math, 13 (2011), 697-725.
doi: 10.1142/S0219199711004282. |
[3] |
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.
doi: 10.1007/s005260050080. |
[4] |
S. Baraket and D. Ye,
Singular limit solutions for two-dimentional elliptic problems with exponentionally dominated nonlinearity, Chinese Ann. Math. Ser. B, 22 (2001), 287-296.
doi: 10.1142/S0252959901000309. |
[5] |
S. Baraket, M. Dammak, T. Ouni and F. Pacard,
Singular limits for a 4-dimensional semilinear elliptic problem with exponential nonlinearity, Ann. I. H. Poincaré - AN, 24 (2007), 875-895.
doi: 10.1016/j.anihpc.2006.06.009. |
[6] |
D. Bartolucci and G. Tarantello,
Liouville type equations with singular data and their applications to periodic multivortices for the Electroweak Theory, Comm. Math. Phys, 229 (2002), 3-47.
doi: 10.1007/s002200200664. |
[7] |
W. H. Bonnett,
Magntically self-focussing streams, Phys. Rev, 45 (1934), 890-897.
|
[8] |
S. Chanillo and M. K. H. Kiessling,
Conformaly invariant systems of nonlinear PDE of Liouville type, Geom. Funct. Anal, 5 (1995), 924-947.
doi: 10.1007/BF01902215. |
[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] |
Z. Chen and W. Zou,
On coupled systems of Schrödinger equations, Adv.Differ. Equ, 16 (2011), 775-800.
|
[11] |
Z. Chen and W. Zou,
An optimal constant for the existence of least energy solutions of a coupled Schrödinger system, Calc. Var. Partial Differ. Equ, 48 (2013), 695-711.
doi: 10.1007/s00526-012-0568-2. |
[12] |
Z. Chen and W. Zou,
On linearly coupled Schrödinger systems, Proc.Am. Math. Soc, 142 (2014), 323-333.
doi: 10.1090/S0002-9939-2013-12000-9. |
[13] |
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. |
[14] |
P. Esposito,
Blow up solutions for a Liouville equation with singular data, SIAM J. Math. Anal, 36 (2005), 1310-1345.
doi: 10.1137/S0036141003430548. |
[15] |
B. D. Esry, C. H. Greene, J. P. Burke and J. L. Bohn Jr.,
Hartree-Fock theory for double condensates, Phys. Rev. Lett, 78 (1997), 3594-3597.
doi: 10.1103/PhysRevLett.78.3594. |
[16] |
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. |
[17] |
C.-S. Lin and G. Tarantello,
When blow-up does not imply concentration: A detour from Brézis-Merle's result, Comptes Rendus Mathematique, 354 (2016), 493-498.
doi: 10.1016/j.crma.2016.01.014. |
[18] |
J. Liouville,
Sur l'équation aux différences partielles $\partial^{2}\log\frac{\lambda}{\partial u \partial v}\pm\frac{\lambda}{2a^{2}} = 0 $, J. Math, 18 (1853), 17-72.
|
[19] |
J. Liu, Y. Guo and Y. Zhang,
Liouville-type theorems for polyharmonic systems in RN, J. Differential Equations, 225 (2006), 685-709.
doi: 10.1016/j.jde.2005.10.016. |
[20] |
C.-S. Lin, J. C. Wei and D. Ye,
Classifcation and nondegeneracy of SU(n+1) Toda system, Invent. Math, 190 (2012), 169-207.
doi: 10.1007/s00222-012-0378-3. |
[21] |
C.-S. Lin, J. Wei and C. Zhao,
Classification of blow-up limits for SU(3) singular Toda systems, Analysis and PDE, 8 (2015), 807-837.
doi: 10.2140/apde.2015.8.807. |
[22] |
M. Musso, A. Pistoia and J. Wei,
New blow-up phenomena for SU(n+1) Toda system, J. Differential Equations, 260 (2016), 6232-6266.
doi: 10.1016/j.jde.2015.12.036. |
[23] |
N. Nagasaki and T. Suzuki,
Asymptotic analysis for a two dimensional elliptic eigenvalue problem with dominated nonlinearity, Asymptotic Analysis, 3 (1990), 173-188.
|
[24] |
T. Suzuki, Two dimensional Emden-Fowler equation with exponential nonlinearity, Nonlinear Diffusion Equations and Their Equilibrium Statesd, 3, (Gregynog, 1989), Birkäuser Boston, Boston, MA, 7 (1992), 493–512. |
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