    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:
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##### References:
  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  W. Chen, Y. Fang and R. Yang, Semilinear equations involving the fractional Laplacian on domains, preprint,, \arXiv{1309.7499}., ().   Google Scholar  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.  Google Scholar  W. Chen and C. Li, Regularity of solutions for a system of integral equation, Commun. Pure Appl. Anal., 4 (2005), 1-8. Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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  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.  Google Scholar  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.  Google Scholar  Q. Guan, Integration by parts formula for regional fractional Laplacian, Comm. Math. Phys., 266 (2006), 289-329. doi: 10.1007/s00220-006-0054-9.  Google Scholar  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.  Google Scholar  T. Kulczycki, Properties of Green function of symmetric stable processes, Probab. Math. Statist., 17 (1997), 339-364. Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar
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