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An elliptic system and the critical hyperbola
1. | Universidade Estadual de Campinas, Campinas, CEP 13083-970 |
2. | Universidade Federal da Paríba, Departamento de Matemática, João Pessoa-PB, CEP 58051-900, Brazil |
3. | Universidade Estadual de Campinas, IMECC, Departamento de Matemática, Caixa Postal 6065, CEP 13083-970, Campinas, SP |
References:
[1] |
G. Bernard, An inhomogeneous semilinear equation in entire space, J. Differential Equations, 125 (1996), 184-214.
doi: 10.1006/jdeq.1996.0029. |
[2] |
J. Busca and R. Manásevich, A Liouville type theorem for Lane Emden systems, Indiana Univ. Math. J., 51 (2002), 37-51. |
[3] |
G. Caristi, L. D'Ambrosio and E. Mitidieri, Liouville theorems for some nonlinear inequalities, Tr. Mat. Inst. Steklova 260 (2008), Teor. Funkts. i Nelinein. Uravn. v Chastn. Proizvodn., 97-118; translation in Proc. Steklov Inst. Math. 260 (2008), 90-111.
doi: 10.1134/S0081543808010070. |
[4] |
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. |
[5] |
Ph. Clément, D. G. de Figueiredo and E. Mitidieri, Positive solutions of semilinear elliptic systems, Comm. Partial Differential Equations, 17 (1992), 923-940.
doi: 10.1080/03605309208820869. |
[6] |
Q. Dai, Entire positive solutions for inhomogeneous semilinear elliptic systems, Glasg. Math. J., 47 (2005), 97-114.
doi: 10.1017/S0017089504002101. |
[7] |
D. G. de Figueiredo and P. L. Felmer, A Liouville-type theorem for elliptic systems, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 21 (1994), 387-397. |
[8] |
B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations, 6 (1981), 883-901.
doi: 10.1080/03605308108820196. |
[9] |
L. Grafakos, Classical and Modern Fourier Analysis, Pearson Education, Inc., Upper Saddle River, NJ, 2004. |
[10] |
C. Jin and C. Li, Quantitative analysis of some system of integral equations, Cal. Var. PDEs, 26 (2006), 447-457.
doi: 10.1007/s00526-006-0013-5. |
[11] |
L. Ma and B. Liu, Symmetry results for decay solutions of elliptic systems in the whole space, Adv. Math., 225 (2010), 3052-3063.
doi: 10.1016/j.aim.2010.05.022. |
[12] |
E. Mitidieri, A Rellich type identity and applications, Commun. Partial Differential Equations, 18 (1993), 125-151.
doi: 10.1080/03605309308820923. |
[13] |
E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $\mathbb{R}^{N}$, Differential Integral Equations, 9 (1996), 465-479. |
[14] |
S. I. Pokhozhaev, Elliptic problems in $\mathbf{\mathbbR}^N$ with a supercritical exponent of nonlinearity, Mat. Sb., 182 (1991), 467-489. |
[15] |
P. Poláčik, P. Quittner and Ph. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems. I. Elliptic equations and systems, Duke Math. J., 139 (2007), 555-579.
doi: 10.1215/S0012-7094-07-13935-8. |
[16] |
P. Quittner and Ph. Souplet, A priori estimates and existence for elliptic systems via bootstrap in weighted Lebesgue spaces, Arch. Ration. Mech. Anal., 174 (2004), 49-81.
doi: 10.1007/s00205-004-0323-8. |
[17] |
J. Serrin and H. Zou, Existence of positive solutions of the Lane-Emden system, Dedicated to Prof. C. Vinti (Italian) (Perugia, 1996). Atti Sem. Mat. Fis. Univ. Modena, 46 (1998), suppl., 369-380. |
[18] |
J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden systems, Differential Integral Equations, 9 (1996), 635-653. |
[19] |
P. Souplet, The proof of the Lane-Emden conjecture in four space dimensions, Adv. Math., 221 (2009), 1409-1427.
doi: 10.1016/j.aim.2009.02.014. |
[20] |
M. A. S. Souto, A priori estimates and existence of positive solutions of nonlinear cooperative elliptic systems, Differential Integral Equations, 8 (1995), 1245-1258. |
[21] |
R. Van Der Vorst, Variational identities and applications to differential systems, Arch. Rational Mech. Anal., 116 (1991), 375-398.
doi: 10.1007/BF00375674. |
[22] |
H. Zou, Symmetry of ground states for a semilinear elliptic system, Trans. Amer. Math. Soc., 352 (2000), 1217-1245.
doi: 10.1090/S0002-9947-99-02526-X. |
[23] |
H. Zou, Symmetry of positive solutions of $\Delta u+u^p=0$ in $\mathbf{\mathbbR}^n$, J. Differential Equations, 120 (1995), 46-88.
doi: 10.1006/jdeq.1995.1105. |
show all references
References:
[1] |
G. Bernard, An inhomogeneous semilinear equation in entire space, J. Differential Equations, 125 (1996), 184-214.
doi: 10.1006/jdeq.1996.0029. |
[2] |
J. Busca and R. Manásevich, A Liouville type theorem for Lane Emden systems, Indiana Univ. Math. J., 51 (2002), 37-51. |
[3] |
G. Caristi, L. D'Ambrosio and E. Mitidieri, Liouville theorems for some nonlinear inequalities, Tr. Mat. Inst. Steklova 260 (2008), Teor. Funkts. i Nelinein. Uravn. v Chastn. Proizvodn., 97-118; translation in Proc. Steklov Inst. Math. 260 (2008), 90-111.
doi: 10.1134/S0081543808010070. |
[4] |
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. |
[5] |
Ph. Clément, D. G. de Figueiredo and E. Mitidieri, Positive solutions of semilinear elliptic systems, Comm. Partial Differential Equations, 17 (1992), 923-940.
doi: 10.1080/03605309208820869. |
[6] |
Q. Dai, Entire positive solutions for inhomogeneous semilinear elliptic systems, Glasg. Math. J., 47 (2005), 97-114.
doi: 10.1017/S0017089504002101. |
[7] |
D. G. de Figueiredo and P. L. Felmer, A Liouville-type theorem for elliptic systems, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 21 (1994), 387-397. |
[8] |
B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations, 6 (1981), 883-901.
doi: 10.1080/03605308108820196. |
[9] |
L. Grafakos, Classical and Modern Fourier Analysis, Pearson Education, Inc., Upper Saddle River, NJ, 2004. |
[10] |
C. Jin and C. Li, Quantitative analysis of some system of integral equations, Cal. Var. PDEs, 26 (2006), 447-457.
doi: 10.1007/s00526-006-0013-5. |
[11] |
L. Ma and B. Liu, Symmetry results for decay solutions of elliptic systems in the whole space, Adv. Math., 225 (2010), 3052-3063.
doi: 10.1016/j.aim.2010.05.022. |
[12] |
E. Mitidieri, A Rellich type identity and applications, Commun. Partial Differential Equations, 18 (1993), 125-151.
doi: 10.1080/03605309308820923. |
[13] |
E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $\mathbb{R}^{N}$, Differential Integral Equations, 9 (1996), 465-479. |
[14] |
S. I. Pokhozhaev, Elliptic problems in $\mathbf{\mathbbR}^N$ with a supercritical exponent of nonlinearity, Mat. Sb., 182 (1991), 467-489. |
[15] |
P. Poláčik, P. Quittner and Ph. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems. I. Elliptic equations and systems, Duke Math. J., 139 (2007), 555-579.
doi: 10.1215/S0012-7094-07-13935-8. |
[16] |
P. Quittner and Ph. Souplet, A priori estimates and existence for elliptic systems via bootstrap in weighted Lebesgue spaces, Arch. Ration. Mech. Anal., 174 (2004), 49-81.
doi: 10.1007/s00205-004-0323-8. |
[17] |
J. Serrin and H. Zou, Existence of positive solutions of the Lane-Emden system, Dedicated to Prof. C. Vinti (Italian) (Perugia, 1996). Atti Sem. Mat. Fis. Univ. Modena, 46 (1998), suppl., 369-380. |
[18] |
J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden systems, Differential Integral Equations, 9 (1996), 635-653. |
[19] |
P. Souplet, The proof of the Lane-Emden conjecture in four space dimensions, Adv. Math., 221 (2009), 1409-1427.
doi: 10.1016/j.aim.2009.02.014. |
[20] |
M. A. S. Souto, A priori estimates and existence of positive solutions of nonlinear cooperative elliptic systems, Differential Integral Equations, 8 (1995), 1245-1258. |
[21] |
R. Van Der Vorst, Variational identities and applications to differential systems, Arch. Rational Mech. Anal., 116 (1991), 375-398.
doi: 10.1007/BF00375674. |
[22] |
H. Zou, Symmetry of ground states for a semilinear elliptic system, Trans. Amer. Math. Soc., 352 (2000), 1217-1245.
doi: 10.1090/S0002-9947-99-02526-X. |
[23] |
H. Zou, Symmetry of positive solutions of $\Delta u+u^p=0$ in $\mathbf{\mathbbR}^n$, J. Differential Equations, 120 (1995), 46-88.
doi: 10.1006/jdeq.1995.1105. |
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