# American Institute of Mathematical Sciences

November  2015, 14(6): 2335-2362. doi: 10.3934/cpaa.2015.14.2335

## 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

Received  January 2015 Revised  June 2015 Published  September 2015

In this paper, we study the following nonlinear elliptic system \begin{eqnarray} \left\{\begin{array}{ll} (-\Delta)^{\frac{\alpha}{2}}u_i=f_i(u),\ x\in \Omega,\quad i=1,...,m, \\ u_i(x)=0, \quad \quad\quad \ \ x\in \Omega^c,\quad i=1,...,m, \end{array} \right. \end{eqnarray} where $0 < \alpha < 2$ and $\Omega$ is either the unit ball $B_1(0)=\{x\in \mathbb R^n | \|x\| < 1 \}$ or the half space $\mathbb R_+^n = \{x=(x_1,...,x_n)\in \mathbb R^n | x_n > 0\}$. Instead of investigating the pseudo differential system directly, we study an equivalent integral system, i.e., \begin{eqnarray} u_i(x)=\int_{B_1(0)}G_1(x,y)f_i(u(y))dy,\quad x\in B_1(0),\quad i=1,...,m, \end{eqnarray} and \begin{eqnarray} u_i(x)=C_ix_n^{\frac{\alpha}{2}}+\int_{\mathbb{R}_+^n}G_{\infty}(x,y)f_i(u(y))dy,\quad x\in \mathbb{R}_+^n,\quad i=1,...,m, \end{eqnarray} where $C_i$ are non-negative constants, $G_1(x,y)$ is Green's function for $B_1(0)$ and $G_{\infty}(x,y)$ is Green function of $\mathbb R_+^n$. We use the method of moving planes in integral forms to prove the radial symmetry and monotonicity of positive solutions in $B_1(0)$ and non-existence of positive solutions in $\mathbb R_+^n$. Moreover, we also study regularity of positive solutions in $B_1(0)$.
Citation: Chenchen Mou. Nonlinear elliptic systems involving the fractional Laplacian in the unit ball and on a half space. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2335-2362. doi: 10.3934/cpaa.2015.14.2335
##### References:
 [1] X. Cabre and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian,, \emph{Adv. Math.}, 224 (2010), 2052. doi: 10.1016/j.aim.2010.01.025. Google Scholar [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,, \emph{Invent. Math.}, 171 (2008), 425. doi: 10.1007/s00222-007-0086-6. Google Scholar [3] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian,, \emph{Comm. Partial Differential Equations}, 32 (2007), 1245. doi: 10.1080/03605300600987306. Google Scholar [4] W. Chen, Y. Fang and R. Yang, Semilinear equations involving the fractional Laplacian on domains, preprint,, \arXiv{1309.7499}., (). Google Scholar [5] W. Chen, Y. Fang and R. Yang, Liouville theorems involving the fractional Laplacian on a half space,, \emph{Adv. Math.}, 274 (2015), 167. doi: 10.1016/j.aim.2014.12.013. Google Scholar [6] W. Chen and C. Li, Regularity of solutions for a system of integral equation,, \emph{Commun. Pure Appl. Anal.}, 4 (2005), 1. Google Scholar [7] W. Chen and C. Li, Classification of positive solutions for nonlinear differential and integral systems with critical exponents,, \emph{Acta Math. Sci. Ser. B Engl. Ed.}, 29 (2009), 946. doi: 10.1016/S0252-9602(09)60079-5. Google Scholar [8] W. Chen and C. Li, Methods on Nonlinear Elliptic Equations,, AIMS Series on Differential Equations & Dynamical Systems, 4 (2010). doi: 978-1-60133-006-2. Google Scholar [9] W. Chen and C. Li, Radial symmetry of solutions for some integral systems of Wolff type,, \emph{Discrete Contin. Dyn. Syst.}, 30 (2011), 1083. doi: 10.3934/dcds.2011.30.1083. Google Scholar [10] W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations,, \emph{Comm. Partial Differential Equations}, 30 (2005), 59. doi: 10.1081/PDE-200044445. Google Scholar [11] W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation,, \emph{Comm. Pure Appl. Math.}, 59 (2006), 330. doi: 10.1002/cpa.20116. Google Scholar [12] W. Chen, C. Li, L. Zhang and T. Cheng, A Liouville theorem for $\alpha$-harmonic functions in $\mathbb R_+^n$,, \emph{Discrete Contin. Dyn. Syst.}, (). Google Scholar [13] W. Chen and J. Zhu, Radial symmetry and regularity of solutions for poly-harmonic Dirichlet problems,, \emph{J. Math. Anal. Appl.}, 377 (2011), 744. doi: 10.1016/j.jmaa.2010.11.035. Google Scholar [14] Y. Fang and W. Chen, A Liouville type theorem for poly-harmonic Dirichlet problem in a half space,, \emph{Adv. in Math.}, 229 (2012), 2835. doi: 10.1016/j.aim.2012.01.018. Google Scholar [15] Q. Guan, Integration by parts formula for regional fractional Laplacian,, \emph{Comm. Math. Phys.}, 266 (2006), 289. doi: 10.1007/s00220-006-0054-9. Google Scholar [16] Q. Guan and Z. Ma, Reflected symmetric $\alpha$-stable processes and regional fractional Laplacian,, \emph{Probab. Theory Related Fields}, 134 (2006), 649. doi: 10.1007/s00440-005-0438-3. Google Scholar [17] T. Kulczycki, Properties of Green function of symmetric stable processes,, \emph{Probab. Math. Statist.}, 17 (1997), 339. Google Scholar [18] C. Ma, W. Chen and C. Li, Regularity of solutions for an integral system of Wolff type,, \emph{Adv. in Math.}, 3 (2011), 2676. doi: 10.1016/j.aim.2010.07.020. Google Scholar [19] X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplcian: regularity up to the boundary,, \emph{J. Math. Pures Appl.}, 101 (2014), 275. doi: 10.1016/j.matpur.2013.06.003. Google Scholar [20] X. Ros-Oton and J. Serra, The Pohozaev identity for the fractional Laplacian,, \emph{Arch. Ration. Mech. Anal.}, 213 (2014), 723. doi: 10.1007/s00205-014-0740-2. Google Scholar [21] L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator,, \emph{Comm. Pure Appl. Math.}, 60 (2007), 67. doi: 10.1002/cpa.20153. Google Scholar

show all references

##### References:
 [1] X. Cabre and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian,, \emph{Adv. Math.}, 224 (2010), 2052. doi: 10.1016/j.aim.2010.01.025. Google Scholar [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,, \emph{Invent. Math.}, 171 (2008), 425. doi: 10.1007/s00222-007-0086-6. Google Scholar [3] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian,, \emph{Comm. Partial Differential Equations}, 32 (2007), 1245. doi: 10.1080/03605300600987306. Google Scholar [4] W. Chen, Y. Fang and R. Yang, Semilinear equations involving the fractional Laplacian on domains, preprint,, \arXiv{1309.7499}., (). Google Scholar [5] W. Chen, Y. Fang and R. Yang, Liouville theorems involving the fractional Laplacian on a half space,, \emph{Adv. Math.}, 274 (2015), 167. doi: 10.1016/j.aim.2014.12.013. Google Scholar [6] W. Chen and C. Li, Regularity of solutions for a system of integral equation,, \emph{Commun. Pure Appl. Anal.}, 4 (2005), 1. Google Scholar [7] W. Chen and C. Li, Classification of positive solutions for nonlinear differential and integral systems with critical exponents,, \emph{Acta Math. Sci. Ser. B Engl. Ed.}, 29 (2009), 946. doi: 10.1016/S0252-9602(09)60079-5. Google Scholar [8] W. Chen and C. Li, Methods on Nonlinear Elliptic Equations,, AIMS Series on Differential Equations & Dynamical Systems, 4 (2010). doi: 978-1-60133-006-2. Google Scholar [9] W. Chen and C. Li, Radial symmetry of solutions for some integral systems of Wolff type,, \emph{Discrete Contin. Dyn. Syst.}, 30 (2011), 1083. doi: 10.3934/dcds.2011.30.1083. Google Scholar [10] W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations,, \emph{Comm. Partial Differential Equations}, 30 (2005), 59. doi: 10.1081/PDE-200044445. Google Scholar [11] W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation,, \emph{Comm. Pure Appl. Math.}, 59 (2006), 330. doi: 10.1002/cpa.20116. Google Scholar [12] W. Chen, C. Li, L. Zhang and T. Cheng, A Liouville theorem for $\alpha$-harmonic functions in $\mathbb R_+^n$,, \emph{Discrete Contin. Dyn. Syst.}, (). Google Scholar [13] W. Chen and J. Zhu, Radial symmetry and regularity of solutions for poly-harmonic Dirichlet problems,, \emph{J. Math. Anal. Appl.}, 377 (2011), 744. doi: 10.1016/j.jmaa.2010.11.035. Google Scholar [14] Y. Fang and W. Chen, A Liouville type theorem for poly-harmonic Dirichlet problem in a half space,, \emph{Adv. in Math.}, 229 (2012), 2835. doi: 10.1016/j.aim.2012.01.018. Google Scholar [15] Q. Guan, Integration by parts formula for regional fractional Laplacian,, \emph{Comm. Math. Phys.}, 266 (2006), 289. doi: 10.1007/s00220-006-0054-9. Google Scholar [16] Q. Guan and Z. Ma, Reflected symmetric $\alpha$-stable processes and regional fractional Laplacian,, \emph{Probab. Theory Related Fields}, 134 (2006), 649. doi: 10.1007/s00440-005-0438-3. Google Scholar [17] T. Kulczycki, Properties of Green function of symmetric stable processes,, \emph{Probab. Math. Statist.}, 17 (1997), 339. Google Scholar [18] C. Ma, W. Chen and C. Li, Regularity of solutions for an integral system of Wolff type,, \emph{Adv. in Math.}, 3 (2011), 2676. doi: 10.1016/j.aim.2010.07.020. Google Scholar [19] X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplcian: regularity up to the boundary,, \emph{J. Math. Pures Appl.}, 101 (2014), 275. doi: 10.1016/j.matpur.2013.06.003. Google Scholar [20] X. Ros-Oton and J. Serra, The Pohozaev identity for the fractional Laplacian,, \emph{Arch. Ration. Mech. Anal.}, 213 (2014), 723. doi: 10.1007/s00205-014-0740-2. Google Scholar [21] L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator,, \emph{Comm. Pure Appl. Math.}, 60 (2007), 67. doi: 10.1002/cpa.20153. Google Scholar
 [1] Meixia Dou. A direct method of moving planes for fractional Laplacian equations in the unit ball. Communications on Pure & 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 & 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 & Continuous Dynamical Systems - A, 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 & Continuous Dynamical Systems - A, 2018, 38 (9) : 4603-4615. doi: 10.3934/dcds.2018201 [5] Zhenjie Li, Ze Cheng, Dongsheng Li. The Liouville type theorem and local regularity results for nonlinear differential and integral systems. Communications on Pure & Applied Analysis, 2015, 14 (2) : 565-576. doi: 10.3934/cpaa.2015.14.565 [6] Dezhong Chen, Li Ma. A Liouville type Theorem for an integral system. Communications on Pure & Applied Analysis, 2006, 5 (4) : 855-859. doi: 10.3934/cpaa.2006.5.855 [7] Baiyu Liu. Direct method of moving planes for logarithmic Laplacian system in bounded domains. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5339-5349. doi: 10.3934/dcds.2018235 [8] Xiaohui Yu. Liouville type theorem for nonlinear elliptic equation with general nonlinearity. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4947-4966. doi: 10.3934/dcds.2014.34.4947 [9] Pengyan Wang, Pengcheng Niu. A direct method of moving planes for a fully nonlinear nonlocal system. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1707-1718. doi: 10.3934/cpaa.2017082 [10] Pengyan Wang, Pengcheng Niu. Liouville's theorem for a fractional elliptic system. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1545-1558. doi: 10.3934/dcds.2019067 [11] Tomás Sanz-Perela. Regularity of radial stable solutions to semilinear elliptic equations for the fractional Laplacian. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2547-2575. doi: 10.3934/cpaa.2018121 [12] Patricio Felmer, César Torres. Radial symmetry of ground states for a regional fractional Nonlinear Schrödinger Equation. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2395-2406. doi: 10.3934/cpaa.2014.13.2395 [13] Ran Zhuo, Wenxiong Chen, Xuewei Cui, Zixia Yuan. Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 1125-1141. doi: 10.3934/dcds.2016.36.1125 [14] Ran Zhuo, Fengquan Li, Boqiang Lv. Liouville type theorems for Schrödinger system with Navier boundary conditions in a half space. Communications on Pure & Applied Analysis, 2014, 13 (3) : 977-990. doi: 10.3934/cpaa.2014.13.977 [15] Shiren Zhu, Xiaoli Chen, Jianfu Yang. Regularity, symmetry and uniqueness of positive solutions to a nonlinear elliptic system. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2685-2696. doi: 10.3934/cpaa.2013.12.2685 [16] Alberto Farina. Symmetry of components, Liouville-type theorems and classification results for some nonlinear elliptic systems. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5869-5877. doi: 10.3934/dcds.2015.35.5869 [17] Wenxiong Chen, Congming Li. Radial symmetry of solutions for some integral systems of Wolff type. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1083-1093. doi: 10.3934/dcds.2011.30.1083 [18] Yonggang Zhao, Mingxin Wang. An integral equation involving Bessel potentials on half space. Communications on Pure & Applied Analysis, 2015, 14 (2) : 527-548. doi: 10.3934/cpaa.2015.14.527 [19] Genggeng Huang. A Liouville theorem of degenerate elliptic equation and its application. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4549-4566. doi: 10.3934/dcds.2013.33.4549 [20] Tingzhi Cheng. Monotonicity and symmetry of solutions to fractional Laplacian equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3587-3599. doi: 10.3934/dcds.2017154

2018 Impact Factor: 0.925