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February  2020, 40(2): 1013-1063. doi: 10.3934/dcds.2020069

Singular limit solutions for a 2-dimensional semilinear elliptic system of Liouville type in some general case

Sami Baraket 1, , Soumaya Sâanouni 1, and  Nihed Trabelsi 2,,

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

* Corresponding author: Nihed Trabelsi

Received  April 2019 Revised  September 2019 Published  November 2019

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.

Keywords: Liouville type system, critical sobolev exponent, nonlinear domain decomposition method, Pohozaev identity, Scalar curvature.
Mathematics Subject Classification: 35J65, 53C21, 46E35, 65M55, 58J05.
Citation: Sami Baraket, Soumaya Sâanouni, Nihed Trabelsi. Singular limit solutions for a 2-dimensional semilinear elliptic system of Liouville type in some general case. Discrete and Continuous Dynamical Systems, 2020, 40 (2) : 1013-1063. doi: 10.3934/dcds.2020069
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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