Nonlinear elliptic systems involving the fractional Laplacian in the unit ball and on a half space

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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

Abstract
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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 $B_1(0)$.

Keywords: Fractional Laplacian, nonlinear elliptic system, radial symmetry, regularity, half space, method of moving planes, integral equation, unit ball, Liouville type theorem..

Mathematics Subject Classification: Primary: 35J47, 35B06, 35B53; Secondary: 35B6.

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.
[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.
[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.
[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,, \emph{Adv. Math.}, 274 (2015), 167. doi: 10.1016/j.aim.2014.12.013.
[6]	W. Chen and C. Li, Regularity of solutions for a system of integral equation,, \emph{Commun. Pure Appl. Anal.}, 4 (2005), 1.
[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.
[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.
[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.
[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.
[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.
[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.}, ().
[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.
[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.
[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.
[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.
[17]	T. Kulczycki, Properties of Green function of symmetric stable processes,, \emph{Probab. Math. Statist.}, 17 (1997), 339.
[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.
[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.
[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.
[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.

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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.
[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.
[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.
[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,, \emph{Adv. Math.}, 274 (2015), 167. doi: 10.1016/j.aim.2014.12.013.
[6]	W. Chen and C. Li, Regularity of solutions for a system of integral equation,, \emph{Commun. Pure Appl. Anal.}, 4 (2005), 1.
[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.
[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.
[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.
[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.
[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.
[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.}, ().
[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.
[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.
[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.
[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.
[17]	T. Kulczycki, Properties of Green function of symmetric stable processes,, \emph{Probab. Math. Statist.}, 17 (1997), 339.
[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.
[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.
[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.
[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.

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