American Institute of Mathematical Sciences

September  2015, 14(5): 1915-1927. doi: 10.3934/cpaa.2015.14.1915

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

Received  October 2014 Revised  April 2015 Published  June 2015

In this paper, we establish some Liouville type theorems for positive solutions of fractional Hénon equation and system in $\mathbb{R}^n$. First, under some regularity conditions, we show that the above equation and system are equivalent to the some integral equation and system, respectively. Then, we prove Liouville type theorems via the method of moving planes in integral forms.
Citation: Jingbo Dou, Huaiyu Zhou. Liouville theorems for fractional Hénon equation and system on $\mathbb{R}^n$. Communications on Pure and Applied Analysis, 2015, 14 (5) : 1915-1927. doi: 10.3934/cpaa.2015.14.1915
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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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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