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Low regularity well-posedness for Gross-Neveu equations
Liouville theorems for fractional Hénon equation and system on $\mathbb{R}^n$
1. | School of Statistics, Xi'an University of Finance and Economics, Xi'an, Shaanxi, 710100, China |
References:
[1] |
L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Diff. Eqs., 232 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[2] |
S-Y Alice Chang and M. González, Fractional Laplacian in conformal geometry, Adv. Math., 226 (2011), 1410-1432.
doi: 10.1016/j.aim.2010.07.016. |
[3] |
W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations, Comm. Partial Diff. Eqs., 30 (2005), 59-65.
doi: 10.1081/PDE-200044445. |
[4] |
W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343.
doi: 10.1002/cpa.20116. |
[5] |
W. Chen and C. Li, An integral system and the Lane-Emdem conjecture, Disc. Cont. Dyn. Sys., 4 (2009), 1167-1184.
doi: 10.3934/dcds.2009.24.1167. |
[6] |
W. Chen and C. Li, Super polyharmonic property of solutions for PDE systems and its applications, Comm. Pure and Appl. Anal., 12 (2013), 2497-2514.
doi: 10.3934/cpaa.2013.12.2497. |
[7] |
C. Cowan, A Liouville theorem for a fourth order Hénon equation, preprint, arXiv:1110.2246. |
[8] |
J. Dou, Liouville type theorems for the system of integral equations, Appl. Math. Comp., 217 (2010), 2586-2594.
doi: 10.1016/j.amc.2010.07.071. |
[9] |
J. Dou, C. Qu and Y. Han, Symmetry and nonexistence of positive solutions to an integral system with weighted functions, Sci. China Math., 54 (2011), 753-768.
doi: 10.1007/s11425-011-4177-x. |
[10] |
M. Fazly, Liouville theorems for the polyharmonic Hénon-Lane-Emden system, Methods and Appl. Anal., 21 (2014), 265-282.
doi: 10.4310/MAA.2014.v21.n2.a5. |
[11] |
M. Fazly and N. Ghoussoub, On the Hénon-Lane-Emden conjecture, Disc. Cont. Dyn. Sys., 34 (2014), 2513-2533.
doi: 10.3934/dcds.2014.34.2513. |
[12] |
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. |
[13] |
B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbb{R}^{N}$, Math. anal. appl., Part A, 369-402, Adv. in Math. Suppl. Stud., 7a, Academic Press, New York-London, 1981. |
[14] |
Y. Guo and J. Liu, Liouville type theorems for positive solutions of elliptic system in $\mathbb{R}^{N}$, Comm. Partial Differ. Eqs., 33 (2008), 263-284.
doi: 10.1080/03605300701257476. |
[15] |
F. B. 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. |
[16] |
T. Jin, Y. Y. Li and J. Xiong, On a fractional Nirenberg problem, part I: blow up analysis and compactness of solutions, J. Eur. Math. Soc., 16 (2014), 1111-1171.
doi: 10.4171/JEMS/456. |
[17] |
T. Jin, Y. Y. Li and J. Xiong, On a fractional Nirenberg problem, part II: existence of solutions, Int. Math. Res. Notices, 6 (2015), 1555-1589. |
[18] |
C-S. Lin, A classification of solutions of a conformally invariant fourth order equation in $\mathbb{R}^{N}$, Comm. Math. Helv., 73 (1998), 206-231.
doi: 10.1007/s000140050052. |
[19] |
Y. Lei and C. Li, Sharp criteria of Liouville type for some nonlinear systems, preprint, arXiv:1301.6235. |
[20] |
Y. Y. Li, Remark on some conformally invariant integral equations: the method of moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180. |
[21] |
Y. Li and P. Niu, Nonexistence of positive solutions for an integral equation related to Hardy-Sobolev inequality, Act. Appl. Math., 134 (2014), 185-200.
doi: 10.1007/s10440-014-9878-z. |
[22] |
E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118 (1983), 349-374.
doi: 10.2307/2007032. |
[23] |
G. Lu and J. Zhu, Symmetry and regularity of extremals of an integral equation related to the Hardy-Sobolev inequality, Calc. Var. PDE, 42 (2011), 563-577.
doi: 10.1007/s00526-011-0398-7. |
[24] |
E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $\mathbb{R}^N2$, Diff. Inte. Equ., 9 (1996), 465-479. |
[25] |
E. Mitidieri, A Rellich type identity and applications, Comm. Partial Diff. Equ., 18 (1993), 125-151.
doi: 10.1080/03605309308820923. |
[26] |
E. Nezza, G. Palatucci and E. Valdinoci, Hitchhikers guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.
doi: 10.1016/j.bulsci.2011.12.004. |
[27] |
Q. H. Phan and Ph. Souplet, Liouville-type theorems and bounds of solutions of Hardy-Hénon equations, J. Diff. Equ., 252 (2012), 2544-2562.
doi: 10.1016/j.jde.2011.09.022. |
[28] |
L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.
doi: 10.1002/cpa.20153. |
[29] |
J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden systems, Diff. Inte. Equ., 9 (1996), 635-653. |
[30] |
J. Serrin and H. Zou, Existence of positive solutions of the Lane-Emden system, Atti Semin. Mat. Fis. Univ. Modena, 46 (1998), 369-380. |
[31] |
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. |
[32] |
J. Villavert, Sharp existence criteria for positive solutions of Hardy-Sobolev type systems, Comm. Pure and Appl. Anal., 14 (2015), 493-515.
doi: 10.3934/cpaa.2015.14.493. |
[33] |
J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann., 313 (1999), 207-228.
doi: 10.1007/s002080050258. |
[34] |
X. Yu, Liouville type theorems for integral equations and integral systems, Calc. Var. PDE, 46 (2013), 75-95.
doi: 10.1007/s00526-011-0474-z. |
[35] |
C. Zhang, Nonexistence of positive solutions of an integral system with weights, Adv. Diff. Equ., 61 (2011), 1-20. |
[36] |
R. Zhuo, W. Chen, X. Cui and Z. Yuan, A Liouville theorem for the fractional Laplacian and its applications, preprint, arxiv: 1401.7402. |
show all references
References:
[1] |
L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Diff. Eqs., 232 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[2] |
S-Y Alice Chang and M. González, Fractional Laplacian in conformal geometry, Adv. Math., 226 (2011), 1410-1432.
doi: 10.1016/j.aim.2010.07.016. |
[3] |
W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations, Comm. Partial Diff. Eqs., 30 (2005), 59-65.
doi: 10.1081/PDE-200044445. |
[4] |
W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343.
doi: 10.1002/cpa.20116. |
[5] |
W. Chen and C. Li, An integral system and the Lane-Emdem conjecture, Disc. Cont. Dyn. Sys., 4 (2009), 1167-1184.
doi: 10.3934/dcds.2009.24.1167. |
[6] |
W. Chen and C. Li, Super polyharmonic property of solutions for PDE systems and its applications, Comm. Pure and Appl. Anal., 12 (2013), 2497-2514.
doi: 10.3934/cpaa.2013.12.2497. |
[7] |
C. Cowan, A Liouville theorem for a fourth order Hénon equation, preprint, arXiv:1110.2246. |
[8] |
J. Dou, Liouville type theorems for the system of integral equations, Appl. Math. Comp., 217 (2010), 2586-2594.
doi: 10.1016/j.amc.2010.07.071. |
[9] |
J. Dou, C. Qu and Y. Han, Symmetry and nonexistence of positive solutions to an integral system with weighted functions, Sci. China Math., 54 (2011), 753-768.
doi: 10.1007/s11425-011-4177-x. |
[10] |
M. Fazly, Liouville theorems for the polyharmonic Hénon-Lane-Emden system, Methods and Appl. Anal., 21 (2014), 265-282.
doi: 10.4310/MAA.2014.v21.n2.a5. |
[11] |
M. Fazly and N. Ghoussoub, On the Hénon-Lane-Emden conjecture, Disc. Cont. Dyn. Sys., 34 (2014), 2513-2533.
doi: 10.3934/dcds.2014.34.2513. |
[12] |
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. |
[13] |
B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbb{R}^{N}$, Math. anal. appl., Part A, 369-402, Adv. in Math. Suppl. Stud., 7a, Academic Press, New York-London, 1981. |
[14] |
Y. Guo and J. Liu, Liouville type theorems for positive solutions of elliptic system in $\mathbb{R}^{N}$, Comm. Partial Differ. Eqs., 33 (2008), 263-284.
doi: 10.1080/03605300701257476. |
[15] |
F. B. 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. |
[16] |
T. Jin, Y. Y. Li and J. Xiong, On a fractional Nirenberg problem, part I: blow up analysis and compactness of solutions, J. Eur. Math. Soc., 16 (2014), 1111-1171.
doi: 10.4171/JEMS/456. |
[17] |
T. Jin, Y. Y. Li and J. Xiong, On a fractional Nirenberg problem, part II: existence of solutions, Int. Math. Res. Notices, 6 (2015), 1555-1589. |
[18] |
C-S. Lin, A classification of solutions of a conformally invariant fourth order equation in $\mathbb{R}^{N}$, Comm. Math. Helv., 73 (1998), 206-231.
doi: 10.1007/s000140050052. |
[19] |
Y. Lei and C. Li, Sharp criteria of Liouville type for some nonlinear systems, preprint, arXiv:1301.6235. |
[20] |
Y. Y. Li, Remark on some conformally invariant integral equations: the method of moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180. |
[21] |
Y. Li and P. Niu, Nonexistence of positive solutions for an integral equation related to Hardy-Sobolev inequality, Act. Appl. Math., 134 (2014), 185-200.
doi: 10.1007/s10440-014-9878-z. |
[22] |
E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118 (1983), 349-374.
doi: 10.2307/2007032. |
[23] |
G. Lu and J. Zhu, Symmetry and regularity of extremals of an integral equation related to the Hardy-Sobolev inequality, Calc. Var. PDE, 42 (2011), 563-577.
doi: 10.1007/s00526-011-0398-7. |
[24] |
E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $\mathbb{R}^N2$, Diff. Inte. Equ., 9 (1996), 465-479. |
[25] |
E. Mitidieri, A Rellich type identity and applications, Comm. Partial Diff. Equ., 18 (1993), 125-151.
doi: 10.1080/03605309308820923. |
[26] |
E. Nezza, G. Palatucci and E. Valdinoci, Hitchhikers guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.
doi: 10.1016/j.bulsci.2011.12.004. |
[27] |
Q. H. Phan and Ph. Souplet, Liouville-type theorems and bounds of solutions of Hardy-Hénon equations, J. Diff. Equ., 252 (2012), 2544-2562.
doi: 10.1016/j.jde.2011.09.022. |
[28] |
L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.
doi: 10.1002/cpa.20153. |
[29] |
J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden systems, Diff. Inte. Equ., 9 (1996), 635-653. |
[30] |
J. Serrin and H. Zou, Existence of positive solutions of the Lane-Emden system, Atti Semin. Mat. Fis. Univ. Modena, 46 (1998), 369-380. |
[31] |
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. |
[32] |
J. Villavert, Sharp existence criteria for positive solutions of Hardy-Sobolev type systems, Comm. Pure and Appl. Anal., 14 (2015), 493-515.
doi: 10.3934/cpaa.2015.14.493. |
[33] |
J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann., 313 (1999), 207-228.
doi: 10.1007/s002080050258. |
[34] |
X. Yu, Liouville type theorems for integral equations and integral systems, Calc. Var. PDE, 46 (2013), 75-95.
doi: 10.1007/s00526-011-0474-z. |
[35] |
C. Zhang, Nonexistence of positive solutions of an integral system with weights, Adv. Diff. Equ., 61 (2011), 1-20. |
[36] |
R. Zhuo, W. Chen, X. Cui and Z. Yuan, A Liouville theorem for the fractional Laplacian and its applications, preprint, arxiv: 1401.7402. |
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