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Liouville type theorem for nonlinear elliptic equation with general nonlinearity
1. | The Center for China's Overseas Interests, Shenzhen University, Shenzhen Guangdong, 518060, China |
References:
[1] |
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. |
[2] |
W. Chen and C. Li, An integral system and the Lane-Emdem conjecture,, Disc. Cont. Dyn. Sys., 24 (2009), 1167.
doi: 10.3934/dcds.2009.24.1167. |
[3] |
W. Chen and C. Li, Classification of positive solutions for nonlinear differential and integral systems with critical exponents,, Acta Mathematica Scientia, 29 (2009), 949.
doi: 10.1016/S0252-9602(09)60079-5. |
[4] |
W. Chen and C. Li, Methods on Nonlinear Elliptic Equations,, AIMS Book Series, (2010).
|
[5] |
W. Chen and C. Li, Radial symmetry of solutions for some integral systems of Wolff type,, Discrete Contin. Dyn. Syst., 30 (2011), 1083.
doi: 10.3934/dcds.2011.30.1083. |
[6] |
W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation,, Comm. Pure and Appl. Math., 59 (2006), 330.
doi: 10.1002/cpa.20116. |
[7] |
W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations,, Comm. P.D.E., 30 (2005), 59.
doi: 10.1081/PDE-200044445. |
[8] |
M. Chipot, M. Chlebik, M. Fila and I. Shafrir, Existence of positive solutions of a semilinear elliptic equation in $\mathbb R_+^n$ with a nonlinear boundary condition,, J. Math. Anal. Appl., 223 (1998), 429.
doi: 10.1006/jmaa.1998.5958. |
[9] |
L. Damascelli and F. Gladiali, Some nonexistence results for positive solutions of elliptic equations in unbounded domains,, Rev. Mat. Iberoamericana, 20 (2004), 67.
|
[10] |
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.
|
[11] |
D. G. de Figueiredo, P. L. Lions and R. D. Nussbaum, A priori estimates and existence of positive solutions of semilinear elliptic equation,, J. Math. Pures. Appl., 61 (1982), 41.
|
[12] |
B. Gidas and J. Spruk, A priori bounds for positive solutions of nonlinear elliptic equations,, Comm. P.D.E., 6 (1981), 883.
doi: 10.1002/cpa.3160340406. |
[13] |
B. Gidas, W. Ni and L. Nirenberg, Symmetry and related properties via maximum principle,, Commun. Math. Phys., 68 (1979), 209.
doi: 10.1007/BF01221125. |
[14] |
J. Liu, Y. Guo and Y. Zhang, Liouville-type theorems for polyharmonic systems in $\mathbb R^N$,, Journal of Differential Equations, 225 (2006), 685.
doi: 10.1016/j.jde.2005.10.016. |
[15] |
Y. Guo and J. Liu, Liouville type theorems for positive solutions of elliptic system in $\mathbb R^N$,, Comm. P.D.E., 33 (2008), 263.
doi: 10.1080/03605300701257476. |
[16] |
Y. Guo and J. Liu, Liouville-type theorems for polyharmonic equations in $\mathbb R^N$ and in $\mathbb R^N_+$,, Proceedings of the Royal Society of Edinburgh, 138 (2008), 339.
doi: 10.1017/S0308210506000394. |
[17] |
F. Hang, X. Wang and X. Yan, An integral equation in conformal geometry,, Ann. Inst. H. Poincare Anal. Non Lineaire, 26 (2009), 1.
doi: 10.1016/j.anihpc.2007.03.006. |
[18] |
B. Hu, Nonexistence of a positive solution of the Laplace equation with a nonlinear boundary condition,, Differential Integral Equations, 7 (1994), 301.
|
[19] |
B. Hu and H. Yin, The profile near blowup time for solution of the heat equation with a nonlinear boundary condition,, Trans. Amer. Math. Soc., 346 (1994), 117.
doi: 10.1090/S0002-9947-1994-1270664-3. |
[20] |
C. Jin and C. Li, Symmetry of solutions to some systems of integral equations,, Proc. Amer. Math. Soc., 134 (2006), 1661.
doi: 10.1090/S0002-9939-05-08411-X. |
[21] |
C. Li and L. Ma, Uniqueness of positive bound states to Shrödinger systems with critical exponents,, SIAM J. Math. Anal., 40 (2008), 1049.
doi: 10.1137/080712301. |
[22] |
Y. Li and L. Zhang, Liouville-type theorems and harnack-type inequalities for semilinear elliptic equations,, Journal d'Analyse Mathématique, 90 (2003), 27.
doi: 10.1007/BF02786551. |
[23] |
Y. Li and M. Zhu, Uniqueness theorems through the method of moving spheres,, Duke Math. J., 80 (1995), 383.
doi: 10.1215/S0012-7094-95-08016-8. |
[24] |
Y. Lou and M. Zhu, Classifications of nonnegative solutions to some elliptic problems,, Differential Integral Equations, 12 (1999), 601.
|
[25] |
L. Ma and D. Chen, A Liouville type theorem for an integral system,, Comm. Pure and Appl, 5 (2006), 855.
doi: 10.3934/cpaa.2006.5.855. |
[26] |
C. Ma, W. Chen and C. Li, Regularity of solutions for an integral system of Wolff type,, Advances in Mathematics, 226 (2011), 2676.
doi: 10.1016/j.aim.2010.07.020. |
[27] |
L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation,, Arch. Ration. Mech. Anal., 195 (2010), 455.
doi: 10.1007/s00205-008-0208-3. |
[28] |
E. Mitidieri, Nonexistence of positive solutions of semilinear systems in $ R^N$,, Diff. Int. Eq., 9 (1996), 465.
|
[29] |
B. Ou, Positive harmonic functions on the upper half space satisfying a nonlinear boundary condition,, Differential Integral Equations, 9 (1996), 1157.
|
[30] |
J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden system,, Diff. Int. Eq., 9 (1996), 635.
|
[31] |
J. Serrin and H. Zou, Existence of positive solutions of Lane-Emden system,, Atti Sem. Mat. Fis. Univ. Modena. Sippl., 46 (1998), 369. Google Scholar |
[32] |
J. Serrin and H. Zou, Existence of positive entire solutions of elliptic Hamiltonian systems,, Comm. P.D.E., 23 (1998), 577.
doi: 10.1080/03605309808821356. |
[33] |
P. Souplet, The proof of the Lane-Emden conjecture in four space dimensions,, Advances in Mathematics, 221 (2009), 1409.
doi: 10.1016/j.aim.2009.02.014. |
[34] |
S. Terracini, Symmetry properties of positives solutions to some elliptic equations with nonlinear boundary conditions,, Diff. Int. Eq., 8 (1995), 1911.
|
[35] |
S. Terracini, On positive entire solutions to a class of equations with a singular coefficient and critical exponent,, Adv. Diff. Eq., 1 (1996), 241.
|
[36] |
X. Yu, Liouville type theorems for integral equations and integral systems,, Calc. Var., 46 (2013), 75.
doi: 10.1007/s00526-011-0474-z. |
[37] |
X. Yu, Liouville Type Theorems for Singular Integral Equations and Integral Systems,, preprint., (). Google Scholar |
[38] |
X. Yu, Liouville type theorem in the Heisenberg group with general nonlinearity,, Journal of Differential Equations, 254 (2013), 2173.
doi: 10.1016/j.jde.2012.11.021. |
show all references
References:
[1] |
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. |
[2] |
W. Chen and C. Li, An integral system and the Lane-Emdem conjecture,, Disc. Cont. Dyn. Sys., 24 (2009), 1167.
doi: 10.3934/dcds.2009.24.1167. |
[3] |
W. Chen and C. Li, Classification of positive solutions for nonlinear differential and integral systems with critical exponents,, Acta Mathematica Scientia, 29 (2009), 949.
doi: 10.1016/S0252-9602(09)60079-5. |
[4] |
W. Chen and C. Li, Methods on Nonlinear Elliptic Equations,, AIMS Book Series, (2010).
|
[5] |
W. Chen and C. Li, Radial symmetry of solutions for some integral systems of Wolff type,, Discrete Contin. Dyn. Syst., 30 (2011), 1083.
doi: 10.3934/dcds.2011.30.1083. |
[6] |
W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation,, Comm. Pure and Appl. Math., 59 (2006), 330.
doi: 10.1002/cpa.20116. |
[7] |
W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations,, Comm. P.D.E., 30 (2005), 59.
doi: 10.1081/PDE-200044445. |
[8] |
M. Chipot, M. Chlebik, M. Fila and I. Shafrir, Existence of positive solutions of a semilinear elliptic equation in $\mathbb R_+^n$ with a nonlinear boundary condition,, J. Math. Anal. Appl., 223 (1998), 429.
doi: 10.1006/jmaa.1998.5958. |
[9] |
L. Damascelli and F. Gladiali, Some nonexistence results for positive solutions of elliptic equations in unbounded domains,, Rev. Mat. Iberoamericana, 20 (2004), 67.
|
[10] |
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.
|
[11] |
D. G. de Figueiredo, P. L. Lions and R. D. Nussbaum, A priori estimates and existence of positive solutions of semilinear elliptic equation,, J. Math. Pures. Appl., 61 (1982), 41.
|
[12] |
B. Gidas and J. Spruk, A priori bounds for positive solutions of nonlinear elliptic equations,, Comm. P.D.E., 6 (1981), 883.
doi: 10.1002/cpa.3160340406. |
[13] |
B. Gidas, W. Ni and L. Nirenberg, Symmetry and related properties via maximum principle,, Commun. Math. Phys., 68 (1979), 209.
doi: 10.1007/BF01221125. |
[14] |
J. Liu, Y. Guo and Y. Zhang, Liouville-type theorems for polyharmonic systems in $\mathbb R^N$,, Journal of Differential Equations, 225 (2006), 685.
doi: 10.1016/j.jde.2005.10.016. |
[15] |
Y. Guo and J. Liu, Liouville type theorems for positive solutions of elliptic system in $\mathbb R^N$,, Comm. P.D.E., 33 (2008), 263.
doi: 10.1080/03605300701257476. |
[16] |
Y. Guo and J. Liu, Liouville-type theorems for polyharmonic equations in $\mathbb R^N$ and in $\mathbb R^N_+$,, Proceedings of the Royal Society of Edinburgh, 138 (2008), 339.
doi: 10.1017/S0308210506000394. |
[17] |
F. Hang, X. Wang and X. Yan, An integral equation in conformal geometry,, Ann. Inst. H. Poincare Anal. Non Lineaire, 26 (2009), 1.
doi: 10.1016/j.anihpc.2007.03.006. |
[18] |
B. Hu, Nonexistence of a positive solution of the Laplace equation with a nonlinear boundary condition,, Differential Integral Equations, 7 (1994), 301.
|
[19] |
B. Hu and H. Yin, The profile near blowup time for solution of the heat equation with a nonlinear boundary condition,, Trans. Amer. Math. Soc., 346 (1994), 117.
doi: 10.1090/S0002-9947-1994-1270664-3. |
[20] |
C. Jin and C. Li, Symmetry of solutions to some systems of integral equations,, Proc. Amer. Math. Soc., 134 (2006), 1661.
doi: 10.1090/S0002-9939-05-08411-X. |
[21] |
C. Li and L. Ma, Uniqueness of positive bound states to Shrödinger systems with critical exponents,, SIAM J. Math. Anal., 40 (2008), 1049.
doi: 10.1137/080712301. |
[22] |
Y. Li and L. Zhang, Liouville-type theorems and harnack-type inequalities for semilinear elliptic equations,, Journal d'Analyse Mathématique, 90 (2003), 27.
doi: 10.1007/BF02786551. |
[23] |
Y. Li and M. Zhu, Uniqueness theorems through the method of moving spheres,, Duke Math. J., 80 (1995), 383.
doi: 10.1215/S0012-7094-95-08016-8. |
[24] |
Y. Lou and M. Zhu, Classifications of nonnegative solutions to some elliptic problems,, Differential Integral Equations, 12 (1999), 601.
|
[25] |
L. Ma and D. Chen, A Liouville type theorem for an integral system,, Comm. Pure and Appl, 5 (2006), 855.
doi: 10.3934/cpaa.2006.5.855. |
[26] |
C. Ma, W. Chen and C. Li, Regularity of solutions for an integral system of Wolff type,, Advances in Mathematics, 226 (2011), 2676.
doi: 10.1016/j.aim.2010.07.020. |
[27] |
L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation,, Arch. Ration. Mech. Anal., 195 (2010), 455.
doi: 10.1007/s00205-008-0208-3. |
[28] |
E. Mitidieri, Nonexistence of positive solutions of semilinear systems in $ R^N$,, Diff. Int. Eq., 9 (1996), 465.
|
[29] |
B. Ou, Positive harmonic functions on the upper half space satisfying a nonlinear boundary condition,, Differential Integral Equations, 9 (1996), 1157.
|
[30] |
J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden system,, Diff. Int. Eq., 9 (1996), 635.
|
[31] |
J. Serrin and H. Zou, Existence of positive solutions of Lane-Emden system,, Atti Sem. Mat. Fis. Univ. Modena. Sippl., 46 (1998), 369. Google Scholar |
[32] |
J. Serrin and H. Zou, Existence of positive entire solutions of elliptic Hamiltonian systems,, Comm. P.D.E., 23 (1998), 577.
doi: 10.1080/03605309808821356. |
[33] |
P. Souplet, The proof of the Lane-Emden conjecture in four space dimensions,, Advances in Mathematics, 221 (2009), 1409.
doi: 10.1016/j.aim.2009.02.014. |
[34] |
S. Terracini, Symmetry properties of positives solutions to some elliptic equations with nonlinear boundary conditions,, Diff. Int. Eq., 8 (1995), 1911.
|
[35] |
S. Terracini, On positive entire solutions to a class of equations with a singular coefficient and critical exponent,, Adv. Diff. Eq., 1 (1996), 241.
|
[36] |
X. Yu, Liouville type theorems for integral equations and integral systems,, Calc. Var., 46 (2013), 75.
doi: 10.1007/s00526-011-0474-z. |
[37] |
X. Yu, Liouville Type Theorems for Singular Integral Equations and Integral Systems,, preprint., (). Google Scholar |
[38] |
X. Yu, Liouville type theorem in the Heisenberg group with general nonlinearity,, Journal of Differential Equations, 254 (2013), 2173.
doi: 10.1016/j.jde.2012.11.021. |
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