Classification of radial solutions to Liouville systems with singularities

Previous Article
Asymptotic symmetry for a class of nonlinear fractional reaction-diffusion equations
DCDS Home
This Issue
Next Article
Rigidity results for nonlocal phase transitions in the Heisenberg group $\mathbb{H}$

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

Abstract
Related Papers

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

[1]	Pedro J. Torres, Zhibo Cheng, Jingli Ren. Non-degeneracy and uniqueness of periodic solutions for $2n$-order differential equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 2155-2168. doi: 10.3934/dcds.2013.33.2155
[2]	Robert Magnus, Olivier Moschetta. The non-linear Schrödinger equation with non-periodic potential: infinite-bump solutions and non-degeneracy. Communications on Pure & Applied Analysis, 2012, 11 (2) : 587-626. doi: 10.3934/cpaa.2012.11.587
[3]	Xiaocai Wang, Junxiang Xu. Gevrey-smoothness of invariant tori for analytic reversible systems under Rüssmann's non-degeneracy condition. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 701-718. doi: 10.3934/dcds.2009.25.701
[4]	Yingshu Lü. Symmetry and non-existence of solutions to an integral system. Communications on Pure & Applied Analysis, 2018, 17 (3) : 807-821. doi: 10.3934/cpaa.2018041
[5]	Dongfeng Zhang, Junxiang Xu. On elliptic lower dimensional tori for Gevrey-smooth Hamiltonian systems under Rüssmann's non-degeneracy condition. Discrete & Continuous Dynamical Systems - A, 2006, 16 (3) : 635-655. doi: 10.3934/dcds.2006.16.635
[6]	Ran Zhuo, Wenxiong Chen, Xuewei Cui, Zixia Yuan. Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 1125-1141. doi: 10.3934/dcds.2016.36.1125
[7]	Antonio Greco, Vincenzino Mascia. Non-local sublinear problems: Existence, comparison, and radial symmetry. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 503-519. doi: 10.3934/dcds.2019021
[8]	Alberto Farina. Symmetry of components, Liouville-type theorems and classification results for some nonlinear elliptic systems. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5869-5877. doi: 10.3934/dcds.2015.35.5869
[9]	Olivier Goubet. Regularity of extremal solutions of a Liouville system. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 339-345. doi: 10.3934/dcdss.2019023
[10]	Wenxiong Chen, Congming Li. Radial symmetry of solutions for some integral systems of Wolff type. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1083-1093. doi: 10.3934/dcds.2011.30.1083
[11]	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
[12]	Jérôme Coville, Juan Dávila. Existence of radial stationary solutions for a system in combustion theory. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 739-766. doi: 10.3934/dcdsb.2011.16.739
[13]	Yuxia Guo, Jianjun Nie. Classification for positive solutions of degenerate elliptic system. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1457-1475. doi: 10.3934/dcds.2018130
[14]	Jingbo Dou, Huaiyu Zhou. Liouville theorems for fractional Hénon equation and system on $\mathbb{R}^n$. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1915-1927. doi: 10.3934/cpaa.2015.14.1915
[15]	Marco Ghimenti, A. M. Micheletti. Non degeneracy for solutions of singularly perturbed nonlinear elliptic problems on symmetric Riemannian manifolds. Communications on Pure & Applied Analysis, 2013, 12 (2) : 679-693. doi: 10.3934/cpaa.2013.12.679
[16]	Rossella Bartolo, Anna Maria Candela, Addolorata Salvatore. Infinitely many radial solutions of a non--homogeneous $p$--Laplacian problem. Conference Publications, 2013, 2013 (special) : 51-59. doi: 10.3934/proc.2013.2013.51
[17]	Myoungjean Bae, Yong Park. Radial transonic shock solutions of Euler-Poisson system in convergent nozzles. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : 773-791. doi: 10.3934/dcdss.2018049
[18]	Hatem Hajlaoui, Abdellaziz Harrabi, Foued Mtiri. Liouville theorems for stable solutions of the weighted Lane-Emden system. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 265-279. doi: 10.3934/dcds.2017011
[19]	E. N. Dancer, Sanjiban Santra. Existence and multiplicity of solutions for a weakly coupled radial system in a ball. Communications on Pure & Applied Analysis, 2008, 7 (4) : 787-793. doi: 10.3934/cpaa.2008.7.787
[20]	Chia-Yu Hsieh. Stability of radial solutions of the Poisson-Nernst-Planck system in annular domains. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2657-2681. doi: 10.3934/dcdsb.2018269

2018 Impact Factor: 1.143

American Institute of Mathematical Sciences