American Institute of Mathematical Sciences

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

Classification of radial solutions to Liouville systems with singularities

 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.
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:
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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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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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