-
Previous Article
Harnack inequality for degenerate elliptic equations and sum operators
- CPAA Home
- This Issue
-
Next Article
Krasnosel'skii type formula and translation along trajectories method on the scale of fractional spaces
Nonlinear elliptic systems involving the fractional Laplacian in the unit ball and on a half space
1. | School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, USA, United States |
References:
[1] |
X. Cabre and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 224 (2010), 2052-2093.
doi: 10.1016/j.aim.2010.01.025. |
[2] |
L. Caffarelli, S. Salsa and L. Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. Math., 171 (2008), 425-461.
doi: 10.1007/s00222-007-0086-6. |
[3] |
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. |
[4] |
W. Chen, Y. Fang and R. Yang, Semilinear equations involving the fractional Laplacian on domains, preprint, arXiv:1309.7499. |
[5] |
W. Chen, Y. Fang and R. Yang, Liouville theorems involving the fractional Laplacian on a half space, Adv. Math., 274 (2015), 167-198.
doi: 10.1016/j.aim.2014.12.013. |
[6] |
W. Chen and C. Li, Regularity of solutions for a system of integral equation, Commun. Pure Appl. Anal., 4 (2005), 1-8. |
[7] |
W. Chen and C. Li, Classification of positive solutions for nonlinear differential and integral systems with critical exponents, Acta Math. Sci. Ser. B Engl. Ed., 29 (2009), 946-960.
doi: 10.1016/S0252-9602(09)60079-5. |
[8] |
W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, AIMS Series on Differential Equations & Dynamical Systems, 4. AIMS, Springfield, MO, 2010.
doi: 978-1-60133-006-2. |
[9] |
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. |
[10] |
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. |
[11] |
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. |
[12] |
W. Chen, C. Li, L. Zhang and T. Cheng, A Liouville theorem for $\alpha$-harmonic functions in $\mathbb R_+^n$, Discrete Contin. Dyn. Syst., in press. |
[13] |
W. Chen and J. Zhu, Radial symmetry and regularity of solutions for poly-harmonic Dirichlet problems, J. Math. Anal. Appl., 377 (2011), 744-753.
doi: 10.1016/j.jmaa.2010.11.035. |
[14] |
Y. Fang and W. Chen, A Liouville type theorem for poly-harmonic Dirichlet problem in a half space, Adv. in Math., 229 (2012), 2835-2867.
doi: 10.1016/j.aim.2012.01.018. |
[15] |
Q. Guan, Integration by parts formula for regional fractional Laplacian, Comm. Math. Phys., 266 (2006), 289-329.
doi: 10.1007/s00220-006-0054-9. |
[16] |
Q. Guan and Z. Ma, Reflected symmetric $\alpha$-stable processes and regional fractional Laplacian, Probab. Theory Related Fields, 134 (2006), 649-694.
doi: 10.1007/s00440-005-0438-3. |
[17] |
T. Kulczycki, Properties of Green function of symmetric stable processes, Probab. Math. Statist., 17 (1997), 339-364. |
[18] |
C. Ma, W. Chen and C. Li, Regularity of solutions for an integral system of Wolff type, Adv. in Math., 3 (2011), 2676-2699.
doi: 10.1016/j.aim.2010.07.020. |
[19] |
X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplcian: regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 275-302.
doi: 10.1016/j.matpur.2013.06.003. |
[20] |
X. Ros-Oton and J. Serra, The Pohozaev identity for the fractional Laplacian, Arch. Ration. Mech. Anal., 213 (2014), 723-750.
doi: 10.1007/s00205-014-0740-2. |
[21] |
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. |
show all references
References:
[1] |
X. Cabre and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 224 (2010), 2052-2093.
doi: 10.1016/j.aim.2010.01.025. |
[2] |
L. Caffarelli, S. Salsa and L. Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. Math., 171 (2008), 425-461.
doi: 10.1007/s00222-007-0086-6. |
[3] |
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. |
[4] |
W. Chen, Y. Fang and R. Yang, Semilinear equations involving the fractional Laplacian on domains, preprint, arXiv:1309.7499. |
[5] |
W. Chen, Y. Fang and R. Yang, Liouville theorems involving the fractional Laplacian on a half space, Adv. Math., 274 (2015), 167-198.
doi: 10.1016/j.aim.2014.12.013. |
[6] |
W. Chen and C. Li, Regularity of solutions for a system of integral equation, Commun. Pure Appl. Anal., 4 (2005), 1-8. |
[7] |
W. Chen and C. Li, Classification of positive solutions for nonlinear differential and integral systems with critical exponents, Acta Math. Sci. Ser. B Engl. Ed., 29 (2009), 946-960.
doi: 10.1016/S0252-9602(09)60079-5. |
[8] |
W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, AIMS Series on Differential Equations & Dynamical Systems, 4. AIMS, Springfield, MO, 2010.
doi: 978-1-60133-006-2. |
[9] |
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. |
[10] |
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. |
[11] |
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. |
[12] |
W. Chen, C. Li, L. Zhang and T. Cheng, A Liouville theorem for $\alpha$-harmonic functions in $\mathbb R_+^n$, Discrete Contin. Dyn. Syst., in press. |
[13] |
W. Chen and J. Zhu, Radial symmetry and regularity of solutions for poly-harmonic Dirichlet problems, J. Math. Anal. Appl., 377 (2011), 744-753.
doi: 10.1016/j.jmaa.2010.11.035. |
[14] |
Y. Fang and W. Chen, A Liouville type theorem for poly-harmonic Dirichlet problem in a half space, Adv. in Math., 229 (2012), 2835-2867.
doi: 10.1016/j.aim.2012.01.018. |
[15] |
Q. Guan, Integration by parts formula for regional fractional Laplacian, Comm. Math. Phys., 266 (2006), 289-329.
doi: 10.1007/s00220-006-0054-9. |
[16] |
Q. Guan and Z. Ma, Reflected symmetric $\alpha$-stable processes and regional fractional Laplacian, Probab. Theory Related Fields, 134 (2006), 649-694.
doi: 10.1007/s00440-005-0438-3. |
[17] |
T. Kulczycki, Properties of Green function of symmetric stable processes, Probab. Math. Statist., 17 (1997), 339-364. |
[18] |
C. Ma, W. Chen and C. Li, Regularity of solutions for an integral system of Wolff type, Adv. in Math., 3 (2011), 2676-2699.
doi: 10.1016/j.aim.2010.07.020. |
[19] |
X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplcian: regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 275-302.
doi: 10.1016/j.matpur.2013.06.003. |
[20] |
X. Ros-Oton and J. Serra, The Pohozaev identity for the fractional Laplacian, Arch. Ration. Mech. Anal., 213 (2014), 723-750.
doi: 10.1007/s00205-014-0740-2. |
[21] |
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. |
[1] |
Meixia Dou. A direct method of moving planes for fractional Laplacian equations in the unit ball. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1797-1807. doi: 10.3934/cpaa.2016015 |
[2] |
Weiwei Zhao, Jinge Yang, Sining Zheng. Liouville type theorem to an integral system in the half-space. Communications on Pure and Applied Analysis, 2014, 13 (2) : 511-525. doi: 10.3934/cpaa.2014.13.511 |
[3] |
Jingbo Dou, Ye Li. Liouville theorem for an integral system on the upper half space. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 155-171. doi: 10.3934/dcds.2015.35.155 |
[4] |
Miaomiao Cai, Li Ma. Moving planes for nonlinear fractional Laplacian equation with negative powers. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4603-4615. doi: 10.3934/dcds.2018201 |
[5] |
Xinjing Wang. Liouville type theorem for Fractional Laplacian system. Communications on Pure and Applied Analysis, 2020, 19 (11) : 5253-5268. doi: 10.3934/cpaa.2020236 |
[6] |
Xiaomei Chen, Xiaohui Yu. Liouville type theorem for Hartree-Fock Equation on half space. Communications on Pure and Applied Analysis, 2022, 21 (6) : 2079-2100. doi: 10.3934/cpaa.2022050 |
[7] |
Zhenjie Li, Ze Cheng, Dongsheng Li. The Liouville type theorem and local regularity results for nonlinear differential and integral systems. Communications on Pure and Applied Analysis, 2015, 14 (2) : 565-576. doi: 10.3934/cpaa.2015.14.565 |
[8] |
Dezhong Chen, Li Ma. A Liouville type Theorem for an integral system. Communications on Pure and Applied Analysis, 2006, 5 (4) : 855-859. doi: 10.3934/cpaa.2006.5.855 |
[9] |
Baiyu Liu. Direct method of moving planes for logarithmic Laplacian system in bounded domains. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 5339-5349. doi: 10.3934/dcds.2018235 |
[10] |
Xiaohui Yu. Liouville type theorem for nonlinear elliptic equation with general nonlinearity. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4947-4966. doi: 10.3934/dcds.2014.34.4947 |
[11] |
Pengyan Wang, Pengcheng Niu. A direct method of moving planes for a fully nonlinear nonlocal system. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1707-1718. doi: 10.3934/cpaa.2017082 |
[12] |
Pengyan Wang, Pengcheng Niu. Liouville's theorem for a fractional elliptic system. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1545-1558. doi: 10.3934/dcds.2019067 |
[13] |
Ziyi Cai, Haiyang He. Asymptotic behavior of solutions for nonlinear integral equations with Hénon type on the unit Ball. Communications on Pure and Applied Analysis, 2020, 19 (9) : 4349-4362. doi: 10.3934/cpaa.2020196 |
[14] |
Tomás Sanz-Perela. Regularity of radial stable solutions to semilinear elliptic equations for the fractional Laplacian. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2547-2575. doi: 10.3934/cpaa.2018121 |
[15] |
Patricio Felmer, César Torres. Radial symmetry of ground states for a regional fractional Nonlinear Schrödinger Equation. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2395-2406. doi: 10.3934/cpaa.2014.13.2395 |
[16] |
Zhenjie Li, Chunqin Zhou. Radial symmetry of nonnegative solutions for nonlinear integral systems. Communications on Pure and Applied Analysis, 2022, 21 (3) : 837-844. doi: 10.3934/cpaa.2021201 |
[17] |
Yuxia Guo, Shaolong Peng. A direct method of moving planes for fully nonlinear nonlocal operators and applications. Discrete and Continuous Dynamical Systems - S, 2021, 14 (6) : 1871-1897. doi: 10.3934/dcdss.2020462 |
[18] |
Ran Zhuo, Wenxiong Chen, Xuewei Cui, Zixia Yuan. Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 1125-1141. doi: 10.3934/dcds.2016.36.1125 |
[19] |
Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $ p $-Laplacian. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3851-3863. doi: 10.3934/dcdss.2020445 |
[20] |
Shiren Zhu, Xiaoli Chen, Jianfu Yang. Regularity, symmetry and uniqueness of positive solutions to a nonlinear elliptic system. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2685-2696. doi: 10.3934/cpaa.2013.12.2685 |
2021 Impact Factor: 1.273
Tools
Metrics
Other articles
by authors
[Back to Top]