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Singularity formation for relativistic Euler and Euler-Poisson equations with repulsive force
The Liouville type theorem and local regularity results for nonlinear differential and integral systems
1. | School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, 710049, China, China |
2. | Department of Applied Mathematics, University of Colorado at Boulder, Colorado |
References:
[1] |
J. Busca and R. Manásevich, A Liouville-type theorem for Lane-Emden system, Indi. Unve. Math. J, 51 (2002), 37-51.
doi: 10.2307/2152750. |
[2] |
G. Caristi, L. D'Ambrosio and E. Mitidieri, Representation formulae for solutions to some classes of higher order systems and related Liouville theorems, Milan. J. Math., 76 (2008), 27-67.
doi: 10.2307/2152750. |
[3] |
L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure. Appl. Math., 42 (1989), 271-297.
doi: 10.2307/2152750. |
[4] |
W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J, 63 (1991), 615-622.
doi: 10.2307/2152750. |
[5] |
W. Chen and C. Li, Classification of positive solutions for nonlinear differential and integral systems with critical exponents, Acta. Math. Scie, 29B (2009), 949-960.
doi: 10.2307/2152750. |
[6] |
W. Chen and C. Li, Super polyharmonic property of solutions for PDE systems and its applications, Comm. Pure. Appl. Anna, 12 (2013), 2497-2514.
doi: 10.2307/2152750. |
[7] |
W. Chen and C. Li, An integral system and the Lane-Emden conjecture, Disc. Cont. Dyna. Syst-A, 24 (2009), 1167-1184.
doi: 10.2307/2152750. |
[8] |
W Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure. Appl. Math, 59 (2006), 330-343.
doi: 10.2307/2152750. |
[9] |
W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations, Comm. Part. Diff. Equa, 30 (2005), 59-65.
doi: 10.2307/2152750. |
[10] |
W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation, Disc. Cont. Dyna. Syst-A, 12 (2005), 347-354.
doi: 10.2307/2152750. |
[11] |
A. Chang and P. Yang, On uniqueness of an $n$-th order differential equation in conformal geometry, Math. Res. Lett, 4 (1997), 91-102.
doi: 10.2307/2152750. |
[12] |
L. Damascelli and F. Gladiali, Some nonexistence results for positive solutions of elliptic equations in unbounded domains, Rev Mat Iber, 20 (2004), 67-86.
doi: 10.2307/2152750. |
[13] |
P. H. Fowler, Further studies of emden's and similar differential equations, Quar. J. Math (Oxford), 2 (1931), 259-288.
doi: 10.2307/2152750. |
[14] |
D. G. de Figueiredo and P. L. Felmer, A Liouville-type theorem for elliptic systems, Anna. Scuola Norm. Sup. Pisa, 21 (1994), 387-397.
doi: 10.2307/2152750. |
[15] |
Y. Guo and J. Liu, Liouville type theorems for positive solutions of elliptic system in $\mathbbR^n$, Comm. Part. Diff. Equa, 33 (2008), 263-284.
doi: 10.2307/2152750. |
[16] |
B. Gidas, W. Ni and L. Nirenberg, Symmetry of Positive Solutions of Nonlinear Elliptic Equations in $\mathbbR^n$, collected in the book Mathematical Analysis and Applications, which is vol. 7a of the book series Advances in Mathematics. Supplementary Studies, Academic Press, New York, 1981.
doi: 10.1007/978-1-4612-0873-0. |
[17] |
B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure. Appl. Math, 34 (1981), 525-598.
doi: 10.2307/2152750. |
[18] |
F. Hang, On the integral systems related to Hardy-Littlewood-Sobolev inequality, Math Res. Lett, 14 (2007), 373-383.
doi: 10.2307/2152750. |
[19] |
T. Kanna and M. Lakshmanan, Exact soliton solutions, shape changing collisions and partially coherent solitons in coupled nonlinear Schrödinger equations, Phys. Rev. Lett, 86 (2001), 5043-5046.
doi: 10.2307/2152750. |
[20] |
C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Inve. Math, 123 (1996), 221-231.
doi: 10.2307/2152750. |
[21] |
C. Lin, A classification of solutions of a conformally invariant fourth order equation in $\mathbbR^n$, Comm. Math. Helv, 73 (1998), 206-231.
doi: 10.2307/2152750. |
[22] |
J. Liu, Y. Guo and Y. Zhang, Liouville-type theorems for polyharmonic systems in $\mathbbR^n$, J. Diff. Equa, 225 (2006), 685-709.
doi: 10.2307/2152750. |
[23] |
C. Li and L. Ma, Uniqueness of positive bound states to Shrödinger systems with critical exponents, SIAM J. Math. Anal, 40 (2008), 1049-1057.
doi: 10.2307/2152750. |
[24] |
T. Lin and J. Wei, Ground state of N coupled nonlinear Schrödinger equations in $\mathbbR^n$, $n\leq3$, Comm. Math. Phys, 255 (2005), 629-653.
doi: 10.2307/2152750. |
[25] |
T. Lin and J. Wei, Spikes in two coupled nonlinear Schrödinger equations, Ann Inst H Poincaré Anal Non-Lin, 22 (2005), 403-439.
doi: 10.2307/2152750. |
[26] |
E. Mitidieri, Non-existence of positive solutions of semilinear systems in $\mathbbR^n$, Diff. Inte. Equa, 9 (1996), 465-479.
doi: 10.2307/2152750. |
[27] |
C. Ma, W. Chen and C. Li, Regularity of solutions for an integral system of Wolff type, Advances in Math, 226 (2011), 2676-2699.
doi: 10.2307/2152750. |
[28] |
L. Ma and L. Zhao, Sharp thresholds of blow-up and global existence for the coupled nonlinear Schrödinger system, J. Math. phys, 49, (2008), 062103.
doi: 10.2307/2152750. |
[29] |
P. Poláčik, P. Quittner and Ph. Souplet, Singularity and decay estimates in superlinear prooblems via Liouville-type theorems. Part I: elliptic systems, Duke Math. J, 139 (2007), 555-579.
doi: 10.2307/2152750. |
[30] |
W. Reichel and H. Zou, Non-existence results for semilinear cooperative elliptic systems via moving spheres, J. Diff. Equa, 161 (2000), 219-243.
doi: 10.2307/2152750. |
[31] |
Ph. Souplet, The proof of the Lane-Emden conjecture in four space dimensions, Advances in Math, 221 (2009), 1409-1427.
doi: 10.2307/2152750. |
[32] |
E. M. Stein and G. Weiss, Fractional integrals in n-dimensional Euclidean space, J. Math. Mech, 7 (1958), 503-514.
doi: 10.2307/2152750. |
[33] |
J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden system, Diff. Inte. Equa, 9 (1996), 635-653.
doi: 10.2307/2152750. |
[34] |
J. Serrin and H. Zou, Existence of positive solutions of Lane-Emden system, Atti Sem. Mat. Fis. Univ. Modena. Sippl, 46 (1996), 369-380.
doi: 10.2307/2152750. |
[35] |
J. Serrin and H. Zou, The existence of positive entire solutions of elliptic Hamiltonian system, Comm. Part. Diff. Equa, 23 (1998), 577-599.
doi: 10.2307/2152750. |
[36] |
J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations, Math. Anna, 313 (1999), 207-228.
doi: 10.2307/2152750. |
[37] |
X. Yu, Liouville type theorems for integral equations and integral systems, Calc. Vari. Part. Diff. Equa, 46 (2013), 75-95.
doi: 10.2307/2152750. |
show all references
References:
[1] |
J. Busca and R. Manásevich, A Liouville-type theorem for Lane-Emden system, Indi. Unve. Math. J, 51 (2002), 37-51.
doi: 10.2307/2152750. |
[2] |
G. Caristi, L. D'Ambrosio and E. Mitidieri, Representation formulae for solutions to some classes of higher order systems and related Liouville theorems, Milan. J. Math., 76 (2008), 27-67.
doi: 10.2307/2152750. |
[3] |
L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure. Appl. Math., 42 (1989), 271-297.
doi: 10.2307/2152750. |
[4] |
W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J, 63 (1991), 615-622.
doi: 10.2307/2152750. |
[5] |
W. Chen and C. Li, Classification of positive solutions for nonlinear differential and integral systems with critical exponents, Acta. Math. Scie, 29B (2009), 949-960.
doi: 10.2307/2152750. |
[6] |
W. Chen and C. Li, Super polyharmonic property of solutions for PDE systems and its applications, Comm. Pure. Appl. Anna, 12 (2013), 2497-2514.
doi: 10.2307/2152750. |
[7] |
W. Chen and C. Li, An integral system and the Lane-Emden conjecture, Disc. Cont. Dyna. Syst-A, 24 (2009), 1167-1184.
doi: 10.2307/2152750. |
[8] |
W Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure. Appl. Math, 59 (2006), 330-343.
doi: 10.2307/2152750. |
[9] |
W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations, Comm. Part. Diff. Equa, 30 (2005), 59-65.
doi: 10.2307/2152750. |
[10] |
W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation, Disc. Cont. Dyna. Syst-A, 12 (2005), 347-354.
doi: 10.2307/2152750. |
[11] |
A. Chang and P. Yang, On uniqueness of an $n$-th order differential equation in conformal geometry, Math. Res. Lett, 4 (1997), 91-102.
doi: 10.2307/2152750. |
[12] |
L. Damascelli and F. Gladiali, Some nonexistence results for positive solutions of elliptic equations in unbounded domains, Rev Mat Iber, 20 (2004), 67-86.
doi: 10.2307/2152750. |
[13] |
P. H. Fowler, Further studies of emden's and similar differential equations, Quar. J. Math (Oxford), 2 (1931), 259-288.
doi: 10.2307/2152750. |
[14] |
D. G. de Figueiredo and P. L. Felmer, A Liouville-type theorem for elliptic systems, Anna. Scuola Norm. Sup. Pisa, 21 (1994), 387-397.
doi: 10.2307/2152750. |
[15] |
Y. Guo and J. Liu, Liouville type theorems for positive solutions of elliptic system in $\mathbbR^n$, Comm. Part. Diff. Equa, 33 (2008), 263-284.
doi: 10.2307/2152750. |
[16] |
B. Gidas, W. Ni and L. Nirenberg, Symmetry of Positive Solutions of Nonlinear Elliptic Equations in $\mathbbR^n$, collected in the book Mathematical Analysis and Applications, which is vol. 7a of the book series Advances in Mathematics. Supplementary Studies, Academic Press, New York, 1981.
doi: 10.1007/978-1-4612-0873-0. |
[17] |
B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure. Appl. Math, 34 (1981), 525-598.
doi: 10.2307/2152750. |
[18] |
F. Hang, On the integral systems related to Hardy-Littlewood-Sobolev inequality, Math Res. Lett, 14 (2007), 373-383.
doi: 10.2307/2152750. |
[19] |
T. Kanna and M. Lakshmanan, Exact soliton solutions, shape changing collisions and partially coherent solitons in coupled nonlinear Schrödinger equations, Phys. Rev. Lett, 86 (2001), 5043-5046.
doi: 10.2307/2152750. |
[20] |
C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Inve. Math, 123 (1996), 221-231.
doi: 10.2307/2152750. |
[21] |
C. Lin, A classification of solutions of a conformally invariant fourth order equation in $\mathbbR^n$, Comm. Math. Helv, 73 (1998), 206-231.
doi: 10.2307/2152750. |
[22] |
J. Liu, Y. Guo and Y. Zhang, Liouville-type theorems for polyharmonic systems in $\mathbbR^n$, J. Diff. Equa, 225 (2006), 685-709.
doi: 10.2307/2152750. |
[23] |
C. Li and L. Ma, Uniqueness of positive bound states to Shrödinger systems with critical exponents, SIAM J. Math. Anal, 40 (2008), 1049-1057.
doi: 10.2307/2152750. |
[24] |
T. Lin and J. Wei, Ground state of N coupled nonlinear Schrödinger equations in $\mathbbR^n$, $n\leq3$, Comm. Math. Phys, 255 (2005), 629-653.
doi: 10.2307/2152750. |
[25] |
T. Lin and J. Wei, Spikes in two coupled nonlinear Schrödinger equations, Ann Inst H Poincaré Anal Non-Lin, 22 (2005), 403-439.
doi: 10.2307/2152750. |
[26] |
E. Mitidieri, Non-existence of positive solutions of semilinear systems in $\mathbbR^n$, Diff. Inte. Equa, 9 (1996), 465-479.
doi: 10.2307/2152750. |
[27] |
C. Ma, W. Chen and C. Li, Regularity of solutions for an integral system of Wolff type, Advances in Math, 226 (2011), 2676-2699.
doi: 10.2307/2152750. |
[28] |
L. Ma and L. Zhao, Sharp thresholds of blow-up and global existence for the coupled nonlinear Schrödinger system, J. Math. phys, 49, (2008), 062103.
doi: 10.2307/2152750. |
[29] |
P. Poláčik, P. Quittner and Ph. Souplet, Singularity and decay estimates in superlinear prooblems via Liouville-type theorems. Part I: elliptic systems, Duke Math. J, 139 (2007), 555-579.
doi: 10.2307/2152750. |
[30] |
W. Reichel and H. Zou, Non-existence results for semilinear cooperative elliptic systems via moving spheres, J. Diff. Equa, 161 (2000), 219-243.
doi: 10.2307/2152750. |
[31] |
Ph. Souplet, The proof of the Lane-Emden conjecture in four space dimensions, Advances in Math, 221 (2009), 1409-1427.
doi: 10.2307/2152750. |
[32] |
E. M. Stein and G. Weiss, Fractional integrals in n-dimensional Euclidean space, J. Math. Mech, 7 (1958), 503-514.
doi: 10.2307/2152750. |
[33] |
J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden system, Diff. Inte. Equa, 9 (1996), 635-653.
doi: 10.2307/2152750. |
[34] |
J. Serrin and H. Zou, Existence of positive solutions of Lane-Emden system, Atti Sem. Mat. Fis. Univ. Modena. Sippl, 46 (1996), 369-380.
doi: 10.2307/2152750. |
[35] |
J. Serrin and H. Zou, The existence of positive entire solutions of elliptic Hamiltonian system, Comm. Part. Diff. Equa, 23 (1998), 577-599.
doi: 10.2307/2152750. |
[36] |
J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations, Math. Anna, 313 (1999), 207-228.
doi: 10.2307/2152750. |
[37] |
X. Yu, Liouville type theorems for integral equations and integral systems, Calc. Vari. Part. Diff. Equa, 46 (2013), 75-95.
doi: 10.2307/2152750. |
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