American Institute of Mathematical Sciences

Advanced
June  2014, 34(6): 2617-2637. doi: 10.3934/dcds.2014.34.2617

Classification of radial solutions to Liouville systems with singularities

Chang-Shou Lin 1, and  Lei Zhang 2,

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

Received  September 2012 Revised  June 2013 Published  December 2013

Let $A=(a_{ij})_{n\times n}$ be a nonnegative, symmetric, irreducible and invertible matrix. We prove the existence and uniqueness of radial solutions to the following Liouville system with singularity: \begin{eqnarray*} \left\{ \begin{array}{lcl} \Delta u_i+\sum_{j=1}^n a_{ij}|x|^{\beta_j}e^{u_j(x)}=0,\quad \mathbb R^2, \quad i=1,...,n\\ \\ \int_{\mathbb R^2}|x|^{\beta_i}e^{u_i(x)}dx<\infty, \quad i=1,...,n \end{array}\right. \end{eqnarray*} where $\beta_1,...,\beta_n$ are constants greater than $-2$. If all $\beta_i$s are negative we prove that all solutions are radial and the linearized system is non-degenerate.
Keywords: Liouville system, classification of solutions, singularity., radial symmetry, non-degeneracy.
Mathematics Subject Classification: Primary: 35J47; Secondary: 35J6.
Citation: Chang-Shou Lin, Lei Zhang. Classification of radial solutions to Liouville systems with singularities. Discrete and Continuous Dynamical Systems, 2014, 34 (6) : 2617-2637. doi: 10.3934/dcds.2014.34.2617
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.

[1]

Lipeng Duan, Jun Yang. On the non-degeneracy of radial vortex solutions for a coupled Ginzburg-Landau system. Discrete and Continuous Dynamical Systems, 2021, 41 (10) : 4767-4790. doi: 10.3934/dcds.2021056
[2]

Pedro J. Torres, Zhibo Cheng, Jingli Ren. Non-degeneracy and uniqueness of periodic solutions for $2n$-order differential equations. Discrete and Continuous Dynamical Systems, 2013, 33 (5) : 2155-2168. doi: 10.3934/dcds.2013.33.2155
[3]

Robert Magnus, Olivier Moschetta. The non-linear Schrödinger equation with non-periodic potential: infinite-bump solutions and non-degeneracy. Communications on Pure and Applied Analysis, 2012, 11 (2) : 587-626. doi: 10.3934/cpaa.2012.11.587
[4]

Xiaocai Wang, Junxiang Xu. Gevrey-smoothness of invariant tori for analytic reversible systems under Rüssmann's non-degeneracy condition. Discrete and Continuous Dynamical Systems, 2009, 25 (2) : 701-718. doi: 10.3934/dcds.2009.25.701
[5]

Yingshu Lü. Symmetry and non-existence of solutions to an integral system. Communications on Pure and Applied Analysis, 2018, 17 (3) : 807-821. doi: 10.3934/cpaa.2018041
[6]

Dongfeng Zhang, Junxiang Xu. On elliptic lower dimensional tori for Gevrey-smooth Hamiltonian systems under Rüssmann's non-degeneracy condition. Discrete and Continuous Dynamical Systems, 2006, 16 (3) : 635-655. doi: 10.3934/dcds.2006.16.635
[7]

Xia Huang, Liping Wang. Classification to the positive radial solutions with weighted biharmonic equation. Discrete and Continuous Dynamical Systems, 2020, 40 (8) : 4821-4837. doi: 10.3934/dcds.2020203
[8]

Yuhao Yan. Classification of positive radial solutions to a weighted biharmonic equation. Communications on Pure and Applied Analysis, 2021, 20 (12) : 4139-4154. doi: 10.3934/cpaa.2021149
[9]

Zhenjie Li, Chunqin Zhou. Radial symmetry of nonnegative solutions for nonlinear integral systems. Communications on Pure and Applied Analysis, 2022, 21 (3) : 837-844. doi: 10.3934/cpaa.2021201
[10]

Ran Zhuo, Wenxiong Chen, Xuewei Cui, Zixia Yuan. Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 1125-1141. doi: 10.3934/dcds.2016.36.1125
[11]

Teresa D'Aprile. Bubbling solutions for the Liouville equation around a quantized singularity in symmetric domains. Communications on Pure and Applied Analysis, 2021, 20 (1) : 159-191. doi: 10.3934/cpaa.2020262
[12]

Antonio Greco, Vincenzino Mascia. Non-local sublinear problems: Existence, comparison, and radial symmetry. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 503-519. doi: 10.3934/dcds.2019021
[13]

Alberto Farina. Symmetry of components, Liouville-type theorems and classification results for some nonlinear elliptic systems. Discrete and Continuous Dynamical Systems, 2015, 35 (12) : 5869-5877. doi: 10.3934/dcds.2015.35.5869
[14]

Wenxiong Chen, Congming Li. Radial symmetry of solutions for some integral systems of Wolff type. Discrete and Continuous Dynamical Systems, 2011, 30 (4) : 1083-1093. doi: 10.3934/dcds.2011.30.1083
[15]

Sara Barile, Addolorata Salvatore. Radial solutions of semilinear elliptic equations with broken symmetry on unbounded domains. Conference Publications, 2013, 2013 (special) : 41-49. doi: 10.3934/proc.2013.2013.41
[16]

Olivier Goubet. Regularity of extremal solutions of a Liouville system. Discrete and Continuous Dynamical Systems - S, 2019, 12 (2) : 339-345. doi: 10.3934/dcdss.2019023
[17]

Jérôme Coville, Juan Dávila. Existence of radial stationary solutions for a system in combustion theory. Discrete and Continuous Dynamical Systems - B, 2011, 16 (3) : 739-766. doi: 10.3934/dcdsb.2011.16.739
[18]

Yuxia Guo, Jianjun Nie. Classification for positive solutions of degenerate elliptic system. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1457-1475. doi: 10.3934/dcds.2018130
[19]

Alberto Farina, Miguel Angel Navarro. Some Liouville-type results for stable solutions involving the mean curvature operator: The radial case. Discrete and Continuous Dynamical Systems, 2020, 40 (2) : 1233-1256. doi: 10.3934/dcds.2020076
[20]

Marco Ghimenti, A. M. Micheletti. Non degeneracy for solutions of singularly perturbed nonlinear elliptic problems on symmetric Riemannian manifolds. Communications on Pure and Applied Analysis, 2013, 12 (2) : 679-693. doi: 10.3934/cpaa.2013.12.679

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (86)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy