American Institute of Mathematical Sciences

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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 & Continuous Dynamical Systems - A, 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. doi: 10.1081/PDE-200033739. Google Scholar

[2]

W. H. Bennet, Magnetically self-focusing streams,, Phys. Rev., 45 (1934), 890. Google Scholar

[3]

S. Chanillo and M. K.-H. Kiessling, Conformally invariant systems of nonlinear PDE of Liouville type,, Geom. Funct. Anal., 5 (1995), 924. doi: 10.1007/BF01902215. Google Scholar

[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. doi: 10.1007/BF01191617. Google Scholar

[5]

S.-Y. Chang and P. Yang, Prescribing Gaussian curvatuare on S2,, Acta Math., 159 (1987), 215. doi: 10.1007/BF02392560. Google Scholar

[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. Google Scholar

[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. doi: 10.1002/cpa.3014. Google Scholar

[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. doi: 10.3934/dcds.2010.28.1237. Google Scholar

[9]

W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations,, Duke Math. J., 63 (1991), 615. doi: 10.1215/S0012-7094-91-06325-8. Google Scholar

[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. doi: 10.1215/S0012-7094-93-07117-7. Google Scholar

[11]

S. Childress and J. K. Percus, Nonlinear aspects of chemotaxis,, Math. Biosci., 56 (1981), 217. doi: 10.1016/0025-5564(81)90055-9. Google Scholar

[12]

M. Chipot, I. Shafrir and G. Wolansky, On the solutions of Liouville systems,, J. Differential Equations, 140 (1997), 59. doi: 10.1006/jdeq.1997.3316. Google Scholar

[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). Google Scholar

[14]

P. Debye and E. Huckel, Zur theorie der electrolyte,, Phys. Zft, 24 (1923), 305. Google Scholar

[15]

J. Hong, Y. Kim and P. Y. Pac, Multivortex solutions of the abelian Chern-Simons-Higgs theory,, Phys. Rev. Letter, 64 (1990), 2230. doi: 10.1103/PhysRevLett.64.2230. Google Scholar

[16]

R. Jackiw and E. J. Weinberg, Selfdual Chern Simons vortices,, Phys. Rev. Lett., 64 (1990), 2234. doi: 10.1103/PhysRevLett.64.2234. Google Scholar

[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. doi: 10.1002/cpa.20099. Google Scholar

[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. Google Scholar

[19]

J. Jost and G. Wang, Analytic aspects of the Toda system. I. A Moser-Trudinger inequality,, Comm. Pure Appl. Math., 54 (2001), 1289. doi: 10.1002/cpa.10004. Google Scholar

[20]

J. Kazdan and F. Warner, Curvature functions for compact 2-manifolds,, Ann. of Math. (2), 99 (1974), 14. doi: 10.2307/1971012. Google Scholar

[21]

E. F. Keller and L. A. Segel, Traveling bands of Chemotactic Bacteria: A theoretical analysis,, J. Theor. Biol., 30 (1971), 235. doi: 10.1016/0022-5193(71)90051-8. Google Scholar

[22]

M. K.-H. Kiessling and J. L. Lebowitz, Dissipative stationary plasmas: Kinetic modeling, Bennett's pinch and generalizations,, Phys. Plasmas, 1 (1994), 1841. doi: 10.1063/1.870639. Google Scholar

[23]

Y. Y. Li, Harnack type inequality: The method of moving planes,, Comm. Math. Phys., 200 (1999), 421. doi: 10.1007/s002200050536. Google Scholar

[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. doi: 10.1016/j.anihpc.2009.09.001. Google Scholar

[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. doi: 10.1002/cpa.20355. Google Scholar

[26]

M. S. Mock, Asymptotic behavior of solutions of transport equations for semiconductor devices,, J. Math. Anal. Appl., 49 (1975), 215. doi: 10.1016/0022-247X(75)90172-9. Google Scholar

[27]

M. Nolasco and G. Tarantello, Double vortex condensates in the Chern-Simons-Higgs theory,, Calc. Var. and PDE, 9 (1999), 31. doi: 10.1007/s005260050132. Google Scholar

[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. doi: 10.1017/S0308210500001219. Google Scholar

[29]

I. Rubinstein, Electro Diffusion of Ions,, SIAM Studies in Applied Mathematics, (1990). doi: 10.1137/1.9781611970814. Google Scholar

[30]

L. Zhang, Blow up solutions of some nonlinear elliptic equations involving exponential nonlinearities,, Comm. Math. Phys., 268 (2006), 105. doi: 10.1007/s00220-006-0092-3. Google Scholar

[31]

L. Zhang, Asymptotic behavior of blowup solutions for elliptic equations with exponential nonlinearity and singular data,, Commun. Contemp. Math., 11 (2009), 395. doi: 10.1142/S0219199709003417. Google Scholar

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. doi: 10.1081/PDE-200033739. Google Scholar

[2]

W. H. Bennet, Magnetically self-focusing streams,, Phys. Rev., 45 (1934), 890. Google Scholar

[3]

S. Chanillo and M. K.-H. Kiessling, Conformally invariant systems of nonlinear PDE of Liouville type,, Geom. Funct. Anal., 5 (1995), 924. doi: 10.1007/BF01902215. Google Scholar

[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. doi: 10.1007/BF01191617. Google Scholar

[5]

S.-Y. Chang and P. Yang, Prescribing Gaussian curvatuare on S2,, Acta Math., 159 (1987), 215. doi: 10.1007/BF02392560. Google Scholar

[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. Google Scholar

[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. doi: 10.1002/cpa.3014. Google Scholar

[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. doi: 10.3934/dcds.2010.28.1237. Google Scholar

[9]

W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations,, Duke Math. J., 63 (1991), 615. doi: 10.1215/S0012-7094-91-06325-8. Google Scholar

[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. doi: 10.1215/S0012-7094-93-07117-7. Google Scholar

[11]

S. Childress and J. K. Percus, Nonlinear aspects of chemotaxis,, Math. Biosci., 56 (1981), 217. doi: 10.1016/0025-5564(81)90055-9. Google Scholar

[12]

M. Chipot, I. Shafrir and G. Wolansky, On the solutions of Liouville systems,, J. Differential Equations, 140 (1997), 59. doi: 10.1006/jdeq.1997.3316. Google Scholar

[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). Google Scholar

[14]

P. Debye and E. Huckel, Zur theorie der electrolyte,, Phys. Zft, 24 (1923), 305. Google Scholar

[15]

J. Hong, Y. Kim and P. Y. Pac, Multivortex solutions of the abelian Chern-Simons-Higgs theory,, Phys. Rev. Letter, 64 (1990), 2230. doi: 10.1103/PhysRevLett.64.2230. Google Scholar

[16]

R. Jackiw and E. J. Weinberg, Selfdual Chern Simons vortices,, Phys. Rev. Lett., 64 (1990), 2234. doi: 10.1103/PhysRevLett.64.2234. Google Scholar

[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. doi: 10.1002/cpa.20099. Google Scholar

[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. Google Scholar

[19]

J. Jost and G. Wang, Analytic aspects of the Toda system. I. A Moser-Trudinger inequality,, Comm. Pure Appl. Math., 54 (2001), 1289. doi: 10.1002/cpa.10004. Google Scholar

[20]

J. Kazdan and F. Warner, Curvature functions for compact 2-manifolds,, Ann. of Math. (2), 99 (1974), 14. doi: 10.2307/1971012. Google Scholar

[21]

E. F. Keller and L. A. Segel, Traveling bands of Chemotactic Bacteria: A theoretical analysis,, J. Theor. Biol., 30 (1971), 235. doi: 10.1016/0022-5193(71)90051-8. Google Scholar

[22]

M. K.-H. Kiessling and J. L. Lebowitz, Dissipative stationary plasmas: Kinetic modeling, Bennett's pinch and generalizations,, Phys. Plasmas, 1 (1994), 1841. doi: 10.1063/1.870639. Google Scholar

[23]

Y. Y. Li, Harnack type inequality: The method of moving planes,, Comm. Math. Phys., 200 (1999), 421. doi: 10.1007/s002200050536. Google Scholar

[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. doi: 10.1016/j.anihpc.2009.09.001. Google Scholar

[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. doi: 10.1002/cpa.20355. Google Scholar

[26]

M. S. Mock, Asymptotic behavior of solutions of transport equations for semiconductor devices,, J. Math. Anal. Appl., 49 (1975), 215. doi: 10.1016/0022-247X(75)90172-9. Google Scholar

[27]

M. Nolasco and G. Tarantello, Double vortex condensates in the Chern-Simons-Higgs theory,, Calc. Var. and PDE, 9 (1999), 31. doi: 10.1007/s005260050132. Google Scholar

[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. doi: 10.1017/S0308210500001219. Google Scholar

[29]

I. Rubinstein, Electro Diffusion of Ions,, SIAM Studies in Applied Mathematics, (1990). doi: 10.1137/1.9781611970814. Google Scholar

[30]

L. Zhang, Blow up solutions of some nonlinear elliptic equations involving exponential nonlinearities,, Comm. Math. Phys., 268 (2006), 105. doi: 10.1007/s00220-006-0092-3. Google Scholar

[31]

L. Zhang, Asymptotic behavior of blowup solutions for elliptic equations with exponential nonlinearity and singular data,, Commun. Contemp. Math., 11 (2009), 395. doi: 10.1142/S0219199709003417. Google Scholar

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