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

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.

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. BaraketI. Ben OmraneT. 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. BaraketM. DammakT. 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 PinoM. 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. EsryC. H. GreeneJ. 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. LiuY. 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. LinJ. 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. LinJ. 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. MussoA. 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. BaraketI. Ben OmraneT. 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. BaraketM. DammakT. 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 PinoM. 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. EsryC. H. GreeneJ. 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. LiuY. 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. LinJ. 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. LinJ. 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. MussoA. 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.

[1]

Peng Chen, Xiaochun Liu. Multiplicity of solutions to Kirchhoff type equations with critical Sobolev exponent. Communications on Pure and Applied Analysis, 2018, 17 (1) : 113-125. doi: 10.3934/cpaa.2018007

[2]

Y. Kabeya. Behaviors of solutions to a scalar-field equation involving the critical Sobolev exponent with the Robin condition. Discrete and Continuous Dynamical Systems, 2006, 14 (1) : 117-134. doi: 10.3934/dcds.2006.14.117

[3]

T. Ogawa. The degenerate drift-diffusion system with the Sobolev critical exponent. Discrete and Continuous Dynamical Systems - S, 2011, 4 (4) : 875-886. doi: 10.3934/dcdss.2011.4.875

[4]

M. Ben Ayed, Abdelbaki Selmi. Asymptotic behavior and existence results for a biharmonic equation involving the critical Sobolev exponent in a five-dimensional domain. Communications on Pure and Applied Analysis, 2010, 9 (6) : 1705-1722. doi: 10.3934/cpaa.2010.9.1705

[5]

Yanfang Peng. On elliptic systems with Sobolev critical exponent. Discrete and Continuous Dynamical Systems, 2016, 36 (6) : 3357-3373. doi: 10.3934/dcds.2016.36.3357

[6]

Michinori Ishiwata. Existence of a stable set for some nonlinear parabolic equation involving critical Sobolev exponent. Conference Publications, 2005, 2005 (Special) : 443-452. doi: 10.3934/proc.2005.2005.443

[7]

Wenmin Gong, Guangcun Lu. On Dirac equation with a potential and critical Sobolev exponent. Communications on Pure and Applied Analysis, 2015, 14 (6) : 2231-2263. doi: 10.3934/cpaa.2015.14.2231

[8]

Dezhong Chen, Li Ma. A Liouville type Theorem for an integral system. Communications on Pure and Applied Analysis, 2006, 5 (4) : 855-859. doi: 10.3934/cpaa.2006.5.855

[9]

Li Ma. Blow-up for semilinear parabolic equations with critical Sobolev exponent. Communications on Pure and Applied Analysis, 2013, 12 (2) : 1103-1110. doi: 10.3934/cpaa.2013.12.1103

[10]

Xiaomei Sun, Wenyi Chen. Positive solutions for singular elliptic equations with critical Hardy-Sobolev exponent. Communications on Pure and Applied Analysis, 2011, 10 (2) : 527-540. doi: 10.3934/cpaa.2011.10.527

[11]

Kaimin Teng, Xiumei He. Ground state solutions for fractional Schrödinger equations with critical Sobolev exponent. Communications on Pure and Applied Analysis, 2016, 15 (3) : 991-1008. doi: 10.3934/cpaa.2016.15.991

[12]

Guangze Gu, Xianhua Tang, Youpei Zhang. Ground states for asymptotically periodic fractional Kirchhoff equation with critical Sobolev exponent. Communications on Pure and Applied Analysis, 2019, 18 (6) : 3181-3200. doi: 10.3934/cpaa.2019143

[13]

Sitong Chen, Junping Shi, Xianhua Tang. Ground state solutions of Nehari-Pohozaev type for the planar Schrödinger-Poisson system with general nonlinearity. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 5867-5889. doi: 10.3934/dcds.2019257

[14]

Qingping Deng. A nonoverlapping domain decomposition method for nonconforming finite element problems. Communications on Pure and Applied Analysis, 2003, 2 (3) : 297-310. doi: 10.3934/cpaa.2003.2.297

[15]

Jing Xu, Xue-Cheng Tai, Li-Lian Wang. A two-level domain decomposition method for image restoration. Inverse Problems and Imaging, 2010, 4 (3) : 523-545. doi: 10.3934/ipi.2010.4.523

[16]

Yutian Lei, Congming Li. Sharp criteria of Liouville type for some nonlinear systems. Discrete and Continuous Dynamical Systems, 2016, 36 (6) : 3277-3315. doi: 10.3934/dcds.2016.36.3277

[17]

Quoc Hung Phan. Optimal Liouville-type theorems for a parabolic system. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 399-409. doi: 10.3934/dcds.2015.35.399

[18]

Xinjing Wang. Liouville type theorem for Fractional Laplacian system. Communications on Pure and Applied Analysis, 2020, 19 (11) : 5253-5268. doi: 10.3934/cpaa.2020236

[19]

Qi-Lin Xie, Xing-Ping Wu, Chun-Lei Tang. Existence and multiplicity of solutions for Kirchhoff type problem with critical exponent. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2773-2786. doi: 10.3934/cpaa.2013.12.2773

[20]

Qilin Xie, Jianshe Yu. Bounded state solutions of Kirchhoff type problems with a critical exponent in high dimension. Communications on Pure and Applied Analysis, 2019, 18 (1) : 129-158. doi: 10.3934/cpaa.2019008

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (230)
  • HTML views (111)
  • Cited by (1)

Other articles
by authors

[Back to Top]