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1. | Institute of Mathematics, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China |
2. | Department of Applied Mathematical, University of Colorado at Boulder, Boulder, CO 80309, United States |
References:
[1] |
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: 10.1002/cpa.3160420304. |
[2] |
L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[3] |
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: 10.1007/s00032-008-0090-3. |
[4] |
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: 10.4310/MRL.1997.v4.n1.a9. |
[5] |
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. |
[6] |
W. Chen and C. Li, A priori estimates for prescribing scalar curvature equations, Ann. of Math., 145 (1997), 547-564.
doi: 10.2307/2951844. |
[7] |
W. Chen and C. Li, An integral system and the Lane-Emden conjecture, Discrete Contin. Dyn. Syst., 24 (2009), 1167-1184.
doi: 10.3934/dcds.2009.24.1167. |
[8] |
W. Chen and C. Li, Radial symmetry of solutions for some integral systems of Wolff type, Discrete Contin. Dyn. Syst., 30 (2011), 1083-1093.
doi: 10.3934/dcds.2011.30.1083. |
[9] |
W. Chen and C. Li, Super polyharmonic property of solutions for PDE systems and its applications, Commun. Pure Appl. Anal., 12 (2013), 2497-2514.
doi: 10.3934/cpaa.2013.12.2497. |
[10] |
W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation, Discrete Contin. Dyn. Syst., 12 (2005), 347-354.
doi: 10.3934/dcds.2005.12.347. |
[11] |
W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations, Comm. Partial Differential Equations, 30 (2005), 59-65.
doi: 10.1081/PDE-200044445. |
[12] |
W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Commun. Pure Appl. Math., 59 (2006), 330-343.
doi: 10.1002/cpa.20116. |
[13] |
M. Franca, Classification of positive solutions of p-Laplace equation with a growth term, Arch. Math. (Brno), 40 (2004), 415-434. |
[14] |
F. Gazzola, Critical exponents which relate embedding inequalities with quasilinear elliptic operator, in Proceedings of the Fourth International Conference on Dynamical Systems and Differential Equations, May 24-27, 2002, Wilmington, NC, USA, 327-335. |
[15] |
F. Gazzola and H.-C. Grunau, Radial entire solutions for supercritical biharmonic equations, Math. Ann., 334 (2006), 905-936.
doi: 10.1007/s00208-005-0748-x. |
[16] |
B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n$, in Mathematical Analysis and Applications, Part A, Adv. in Math. Suppl. Stud., 7a, Academic Press, New York-London, 1981, 369-402. |
[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: 10.1002/cpa.3160340406. |
[18] |
M. Guedda and L. Veron, Local and global properties of solutions of quasilinear elliptic equations, J. Differential Equations, 76 (1988), 159-189.
doi: 10.1016/0022-0396(88)90068-X. |
[19] |
C. Gui, On positive entire solutions of the elliptic equation $\Delta u+K(x)u^p=0$ and its applications to Riemannian geometry, Proc. Roy. Soc. Edinburgh Sect. A, 126 (1996), 225-237.
doi: 10.1017/S0308210500022708. |
[20] |
F. Hang, On the integral systems related to Hardy-Littlewood-sobolev inequality, Math. Res. Lett., 14 (2007), 373-383.
doi: 10.4310/MRL.2007.v14.n3.a2. |
[21] |
L. I. Hedberg and T. Wolff, Thin sets in nonlinear potential theory, Ann. Inst. Fourier (Grenobel), 33 (1983), 161-187.
doi: 10.5802/aif.944. |
[22] |
C. Jin and C. Li, Qualitative analysis of some systems of integral equations, Calc. Var. Partial Differential Equations, 26 (2006), 447-457.
doi: 10.1007/s00526-006-0013-5. |
[23] |
N. Kawano, E. Yanagida and S. Yotsutani, Structure theorems for positive radial solutions to div$(|Du|^{m-2} Du)+K(|x|)u^q=0$ in $R^n$, J. Math. Soc. Japan, 45 (1993), 719-742.
doi: 10.2969/jmsj/04540719. |
[24] |
T. Kilpelaiinen and J. Malý, The Wiener test and potential estimates for quasilinear elliptic equations, Acta Math., 172 (1994), 137-161.
doi: 10.1007/BF02392793. |
[25] |
D. Labutin, Potential estimates for a class of fully nonlinear elliptic equations, Duke Math. J., 111 (2002), 1-49.
doi: 10.1215/S0012-7094-02-11111-9. |
[26] |
Y. Lei and C. Li, Integrability and asymptotics of positive solutions of a $\gamma$-Laplace system, J. Differential Equations, 252 (2012), 2739-2758.
doi: 10.1016/j.jde.2011.10.009. |
[27] |
Y. Lei, C. Li and C. Ma, Decay estimation for positive solutions of a $\gamma$-Laplace equation, Discrete Contin. Dyn. Syst., 30 (2011), 547-558.
doi: 10.3934/dcds.2011.30.547. |
[28] |
Y. Lei, C. Li and C. Ma, Asymptotic radial symmetry and growth estimates of positive solutions to weighted Hardy-Littlewood-Sobolev system, Calc. Var. Partial Differential Equations, 45 (2012), 43-61.
doi: 10.1007/s00526-011-0450-7. |
[29] |
C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Invent. Math., 123 (1996), 221-231.
doi: 10.1007/s002220050023. |
[30] |
C. Li, A degree theory approach for the shooting method, arXiv:1301.6232v1, 2013. |
[31] |
C. Li and L. Ma, Uniqueness of positive bound states to Schrödinger systems with critical exponents, SIAM J. Math. Anal., 40 (2008), 1049-1057.
doi: 10.1137/080712301. |
[32] |
Y.-Y. Li, Remark on some conformally invariant integral equations: The method of moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180. |
[33] |
Y. Li, Asymptotic behavior of positive solutions of equation $\Delta u+K(x)u^p=0$ in $R^n$, J. Differential Equations, 95 (1992), 304-330.
doi: 10.1016/0022-0396(92)90034-K. |
[34] |
Y. Li and W.-M. Ni, On conformal scalar curvature equations in $R^n$, Duke Math. J., 57 (1988), 895-924.
doi: 10.1215/S0012-7094-88-05740-7. |
[35] |
E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118 (1983), 349-374.
doi: 10.2307/2007032. |
[36] |
C. Lin, A classification of solutions of a conformally invariant fourth order equation in $R^n$, Comm. Math. Helv., 73 (1998), 206-231.
doi: 10.1007/s000140050052. |
[37] |
J. Liu, Y. Guo and Y. Zhang, Liouville-type theorems for polyharmonic systems in $R^n$, J. Differential Equations, 225 (2006), 685-709.
doi: 10.1016/j.jde.2005.10.016. |
[38] |
J. Liu, Y. Guo and Y. Zhang, Existence of positive entire solutions for polyharmonic equations and systems, J. Partial Differential Equations, 19 (2006), 256-270. |
[39] |
C. Ma, W. Chen and C. Li, Regularity of solutions for an integral system of Wolff type, Adv. Math., 226 (2011), 2676-2699.
doi: 10.1016/j.aim.2010.07.020. |
[40] |
E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $R^n$, Differential Integral Equations, 9 (1996), 465-479. |
[41] |
W.-M. Ni and J. Serrin, Existence and nonexistence theorems for ground states of quasilinear partial differential equations. The anomalous case, Accad. Naz. Lincei., 77 (1986), 231-257. |
[42] |
L. Nirenberg, Topics in Nonlinear Functional Analysis, Notes by R. A. Artino, Courant Institute of Mathematical Sciences, New York University, New York, 1974. |
[43] |
M. Otani, Existence and nonexistence of nontrivial solutions of some nonlinear degenerate elliptic equations, J. Funct. Anal., 76 (1988), 140-159.
doi: 10.1016/0022-1236(88)90053-5. |
[44] |
L. A. Peletier and J. Serrin, Ground states for the prescribed mean curvature equation, Proc. Amer. Math. Soc., 100 (1987), 694-700.
doi: 10.1090/S0002-9939-1987-0894440-8. |
[45] |
N. Phuc and I. Verbitsky, Quasilinear and Hessian equations of Lane-Emden type, Ann. of Math., 168 (2008), 859-914.
doi: 10.4007/annals.2008.168.859. |
[46] |
P. Pucci and J. Serrin, A general variational identity, Indiana Univ. Math. J., 35 (1986), 681-703.
doi: 10.1512/iumj.1986.35.35036. |
[47] |
P. Quittner and Ph. Souplet, Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States, Birkhauser Verlag, Basel, 2007. |
[48] |
J. Serrin and H. Zou, Non-existence of positive solution of Lane-Emden systems, Differential Integral Equations, 9 (1996), 635-653. |
[49] |
J. Serrin and H. Zou, Existence of positive solutions of the Lane-Emden system, Atti Sem. Mat. Fis. Univ. Modena, 46 (1998), 369-380. |
[50] |
J. Serrin and H. Zou, Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities, Acta Math.,189 (2002), 79-142.
doi: 10.1007/BF02392645. |
[51] |
Ph. Souplet, The proof of the Lane-Emden conjecture in 4 space dimensions, Adv. Math., 221 (2009), 1409-1427.
doi: 10.1016/j.aim.2009.02.014. |
[52] |
S. Sun and Y. Lei, Fast decay estimates for integrable solutions of the Lane-Emden type integral systems involving the Wolff potentials, J. Funct. Anal., 263 (2012), 3857-3882.
doi: 10.1016/j.jfa.2012.09.012. |
[53] |
J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann., 313 (1999), 207-228.
doi: 10.1007/s002080050258. |
[54] |
X. Yu, Liouville type theorems for integral equations and integral systems, Calc. Var. Partial Differential Equations, 46 (2013), 75-95.
doi: 10.1007/s00526-011-0474-z. |
show all references
References:
[1] |
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: 10.1002/cpa.3160420304. |
[2] |
L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[3] |
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: 10.1007/s00032-008-0090-3. |
[4] |
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: 10.4310/MRL.1997.v4.n1.a9. |
[5] |
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. |
[6] |
W. Chen and C. Li, A priori estimates for prescribing scalar curvature equations, Ann. of Math., 145 (1997), 547-564.
doi: 10.2307/2951844. |
[7] |
W. Chen and C. Li, An integral system and the Lane-Emden conjecture, Discrete Contin. Dyn. Syst., 24 (2009), 1167-1184.
doi: 10.3934/dcds.2009.24.1167. |
[8] |
W. Chen and C. Li, Radial symmetry of solutions for some integral systems of Wolff type, Discrete Contin. Dyn. Syst., 30 (2011), 1083-1093.
doi: 10.3934/dcds.2011.30.1083. |
[9] |
W. Chen and C. Li, Super polyharmonic property of solutions for PDE systems and its applications, Commun. Pure Appl. Anal., 12 (2013), 2497-2514.
doi: 10.3934/cpaa.2013.12.2497. |
[10] |
W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation, Discrete Contin. Dyn. Syst., 12 (2005), 347-354.
doi: 10.3934/dcds.2005.12.347. |
[11] |
W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations, Comm. Partial Differential Equations, 30 (2005), 59-65.
doi: 10.1081/PDE-200044445. |
[12] |
W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Commun. Pure Appl. Math., 59 (2006), 330-343.
doi: 10.1002/cpa.20116. |
[13] |
M. Franca, Classification of positive solutions of p-Laplace equation with a growth term, Arch. Math. (Brno), 40 (2004), 415-434. |
[14] |
F. Gazzola, Critical exponents which relate embedding inequalities with quasilinear elliptic operator, in Proceedings of the Fourth International Conference on Dynamical Systems and Differential Equations, May 24-27, 2002, Wilmington, NC, USA, 327-335. |
[15] |
F. Gazzola and H.-C. Grunau, Radial entire solutions for supercritical biharmonic equations, Math. Ann., 334 (2006), 905-936.
doi: 10.1007/s00208-005-0748-x. |
[16] |
B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n$, in Mathematical Analysis and Applications, Part A, Adv. in Math. Suppl. Stud., 7a, Academic Press, New York-London, 1981, 369-402. |
[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: 10.1002/cpa.3160340406. |
[18] |
M. Guedda and L. Veron, Local and global properties of solutions of quasilinear elliptic equations, J. Differential Equations, 76 (1988), 159-189.
doi: 10.1016/0022-0396(88)90068-X. |
[19] |
C. Gui, On positive entire solutions of the elliptic equation $\Delta u+K(x)u^p=0$ and its applications to Riemannian geometry, Proc. Roy. Soc. Edinburgh Sect. A, 126 (1996), 225-237.
doi: 10.1017/S0308210500022708. |
[20] |
F. Hang, On the integral systems related to Hardy-Littlewood-sobolev inequality, Math. Res. Lett., 14 (2007), 373-383.
doi: 10.4310/MRL.2007.v14.n3.a2. |
[21] |
L. I. Hedberg and T. Wolff, Thin sets in nonlinear potential theory, Ann. Inst. Fourier (Grenobel), 33 (1983), 161-187.
doi: 10.5802/aif.944. |
[22] |
C. Jin and C. Li, Qualitative analysis of some systems of integral equations, Calc. Var. Partial Differential Equations, 26 (2006), 447-457.
doi: 10.1007/s00526-006-0013-5. |
[23] |
N. Kawano, E. Yanagida and S. Yotsutani, Structure theorems for positive radial solutions to div$(|Du|^{m-2} Du)+K(|x|)u^q=0$ in $R^n$, J. Math. Soc. Japan, 45 (1993), 719-742.
doi: 10.2969/jmsj/04540719. |
[24] |
T. Kilpelaiinen and J. Malý, The Wiener test and potential estimates for quasilinear elliptic equations, Acta Math., 172 (1994), 137-161.
doi: 10.1007/BF02392793. |
[25] |
D. Labutin, Potential estimates for a class of fully nonlinear elliptic equations, Duke Math. J., 111 (2002), 1-49.
doi: 10.1215/S0012-7094-02-11111-9. |
[26] |
Y. Lei and C. Li, Integrability and asymptotics of positive solutions of a $\gamma$-Laplace system, J. Differential Equations, 252 (2012), 2739-2758.
doi: 10.1016/j.jde.2011.10.009. |
[27] |
Y. Lei, C. Li and C. Ma, Decay estimation for positive solutions of a $\gamma$-Laplace equation, Discrete Contin. Dyn. Syst., 30 (2011), 547-558.
doi: 10.3934/dcds.2011.30.547. |
[28] |
Y. Lei, C. Li and C. Ma, Asymptotic radial symmetry and growth estimates of positive solutions to weighted Hardy-Littlewood-Sobolev system, Calc. Var. Partial Differential Equations, 45 (2012), 43-61.
doi: 10.1007/s00526-011-0450-7. |
[29] |
C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Invent. Math., 123 (1996), 221-231.
doi: 10.1007/s002220050023. |
[30] |
C. Li, A degree theory approach for the shooting method, arXiv:1301.6232v1, 2013. |
[31] |
C. Li and L. Ma, Uniqueness of positive bound states to Schrödinger systems with critical exponents, SIAM J. Math. Anal., 40 (2008), 1049-1057.
doi: 10.1137/080712301. |
[32] |
Y.-Y. Li, Remark on some conformally invariant integral equations: The method of moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180. |
[33] |
Y. Li, Asymptotic behavior of positive solutions of equation $\Delta u+K(x)u^p=0$ in $R^n$, J. Differential Equations, 95 (1992), 304-330.
doi: 10.1016/0022-0396(92)90034-K. |
[34] |
Y. Li and W.-M. Ni, On conformal scalar curvature equations in $R^n$, Duke Math. J., 57 (1988), 895-924.
doi: 10.1215/S0012-7094-88-05740-7. |
[35] |
E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118 (1983), 349-374.
doi: 10.2307/2007032. |
[36] |
C. Lin, A classification of solutions of a conformally invariant fourth order equation in $R^n$, Comm. Math. Helv., 73 (1998), 206-231.
doi: 10.1007/s000140050052. |
[37] |
J. Liu, Y. Guo and Y. Zhang, Liouville-type theorems for polyharmonic systems in $R^n$, J. Differential Equations, 225 (2006), 685-709.
doi: 10.1016/j.jde.2005.10.016. |
[38] |
J. Liu, Y. Guo and Y. Zhang, Existence of positive entire solutions for polyharmonic equations and systems, J. Partial Differential Equations, 19 (2006), 256-270. |
[39] |
C. Ma, W. Chen and C. Li, Regularity of solutions for an integral system of Wolff type, Adv. Math., 226 (2011), 2676-2699.
doi: 10.1016/j.aim.2010.07.020. |
[40] |
E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $R^n$, Differential Integral Equations, 9 (1996), 465-479. |
[41] |
W.-M. Ni and J. Serrin, Existence and nonexistence theorems for ground states of quasilinear partial differential equations. The anomalous case, Accad. Naz. Lincei., 77 (1986), 231-257. |
[42] |
L. Nirenberg, Topics in Nonlinear Functional Analysis, Notes by R. A. Artino, Courant Institute of Mathematical Sciences, New York University, New York, 1974. |
[43] |
M. Otani, Existence and nonexistence of nontrivial solutions of some nonlinear degenerate elliptic equations, J. Funct. Anal., 76 (1988), 140-159.
doi: 10.1016/0022-1236(88)90053-5. |
[44] |
L. A. Peletier and J. Serrin, Ground states for the prescribed mean curvature equation, Proc. Amer. Math. Soc., 100 (1987), 694-700.
doi: 10.1090/S0002-9939-1987-0894440-8. |
[45] |
N. Phuc and I. Verbitsky, Quasilinear and Hessian equations of Lane-Emden type, Ann. of Math., 168 (2008), 859-914.
doi: 10.4007/annals.2008.168.859. |
[46] |
P. Pucci and J. Serrin, A general variational identity, Indiana Univ. Math. J., 35 (1986), 681-703.
doi: 10.1512/iumj.1986.35.35036. |
[47] |
P. Quittner and Ph. Souplet, Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States, Birkhauser Verlag, Basel, 2007. |
[48] |
J. Serrin and H. Zou, Non-existence of positive solution of Lane-Emden systems, Differential Integral Equations, 9 (1996), 635-653. |
[49] |
J. Serrin and H. Zou, Existence of positive solutions of the Lane-Emden system, Atti Sem. Mat. Fis. Univ. Modena, 46 (1998), 369-380. |
[50] |
J. Serrin and H. Zou, Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities, Acta Math.,189 (2002), 79-142.
doi: 10.1007/BF02392645. |
[51] |
Ph. Souplet, The proof of the Lane-Emden conjecture in 4 space dimensions, Adv. Math., 221 (2009), 1409-1427.
doi: 10.1016/j.aim.2009.02.014. |
[52] |
S. Sun and Y. Lei, Fast decay estimates for integrable solutions of the Lane-Emden type integral systems involving the Wolff potentials, J. Funct. Anal., 263 (2012), 3857-3882.
doi: 10.1016/j.jfa.2012.09.012. |
[53] |
J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann., 313 (1999), 207-228.
doi: 10.1007/s002080050258. |
[54] |
X. Yu, Liouville type theorems for integral equations and integral systems, Calc. Var. Partial Differential Equations, 46 (2013), 75-95.
doi: 10.1007/s00526-011-0474-z. |
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