Regularity and nonexistence of solutions for a system involving the fractional Laplacian

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November 2015, 14(6): 2431-2451. doi: 10.3934/cpaa.2015.14.2431

Regularity and nonexistence of solutions for a system involving the fractional Laplacian

De Tang ^1, and Yanqin Fang ^1,

1.	School of Mathematics, Hunan University, Changsha, 410082, China

Received March 2015 Revised July 2015 Published September 2015

Abstract
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We consider a system involving the fractional Laplacian \begin{eqnarray} \left\{ \begin{array}{ll} (-\Delta)^{\alpha_{1}/2}u=u^{p_{1}}v^{q_{1}} & \mbox{in}\ \mathbb{R}^N_+,\\ (-\Delta)^{\alpha_{2}/2}v=u^{p_{2}}v^{q_{2}} &\mbox{in}\ \mathbb{R}^N_+,\\ u=v=0,&\mbox{in}\ \mathbb{R}^N\backslash\mathbb{R}^N_+, \end{array} \right. \end{eqnarray} where $\alpha_{i}\in (0,2)$, $p_{i},q_{i}>0$, $i=1,2$. Based on the uniqueness of $\alpha$-harmonic function [9] on half space, the equivalence between (1) and integral equations \begin{eqnarray} \left\{ \begin{array}{ll} u(x)=C_{1}x_{N}^{\frac{\alpha_{1}}{2}}+\displaystyle\int_{\mathbb{R}_{+}^{N}}G^{1}_{\infty}(x,y)u^{p_{1}}(y)v^{q_{1}}(y)dy,\\ v(x)=C_{2}x_{N}^{\frac{\alpha_{2}}{2}}+\displaystyle\int_{\mathbb{R}_{+}^{N}}G^{2}_{\infty}(x,y)u^{p_{2}}(y)v^{q_{2}}(y)dy. \end{array} \right. \end{eqnarray} are derived. Based on this result we deal with integral equations (2) instead of (1) and obtain the regularity. Especially, by the method of moving planes in integral forms which is established by Chen-Li-Ou [12], we obtain the nonexistence of positive solutions of integral equations (2) under only local integrability assumptions.

Keywords: Kelvin transform., moving planes in integral forms, nonexistence, integral equations, equivalence, Fractional laplacians.

Mathematics Subject Classification: Primary: 35R11; Secondary: 35A01, 35B5.

Citation: De Tang, Yanqin Fang. Regularity and nonexistence of solutions for a system involving the fractional Laplacian. Communications on Pure and Applied Analysis, 2015, 14 (6) : 2431-2451. doi: 10.3934/cpaa.2015.14.2431

References:

[1]	J. Betoin, Lévy Processes, Cambridge Tracts in Mathematics, 121 Cambridge University Press, Cambridge, 1996.
[2]	H. Berestycki and L. Nirenberg, On the method of moving planes and sliding method, Bol. Soc. Brazil. Mat. (N. S.), 22 (1991), 1-37. doi: 10.1007/BF01244896.
[3]	G. Bianchi, Non-existence of positive solutions to semilinear elliptic equations in $R^N$ and $R_{+}^N$ through the method of moving plane, Comm. PDE., 22 (1997), 1671-1690. doi: 10.1080/03605309708821315.
[4]	K. Bogdan, The boundary Harnack principle for the fractional Laplacian, Studia Math., 123 (1997), 43-80.
[5]	X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates, Ann. I. H. Poincaré-AN., 31 (2014), 23-53. doi: 10.1016/j.anihpc.2013.02.001.
[6]	X. Cabré and J. Solà-Morales, Layer solutions in a half-space for boundary reactions, Comm. Pure Appl. Math., 58 (2005), no.12, 1678-1732. doi: 10.1002/cpa.20093.
[7]	L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. in PDE., 32 (2007), 1245-1260. doi: 10.1080/03605300600987306.
[8]	L. Cao and W. Chen, Liouville type theorems for poly-harmonic Navier problems, Disc. Cont. Dyn. Sys., 33 (2013), 3937-3955. doi: 10.3934/dcds.2013.33.3937.
[9]	W. Chen, Y. Fang and R. Yang, Liouville theorems involving the fractional Laplacian on a half space, Advances in Mathematics, 274 (2015), 167-198. doi: 10.1016/j.aim.2014.12.013.
[10]	W. Chen and C. Li, Regularity of solutions for a system of integral equation, Comm. Pure Appl. Anal., 4 (2005), 1-8.
[11]	W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, AIMS. Ser. Differ. Equ. Dyn. Syst., 4 2010.
[12]	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.
[13]	W. Chen, C. Jin, C. Li and J. Lim, Weighted Hardy-Littlewood-Sobolev inequlaities and systems of integral equations, Discrete Contin. Dyn. Syst., (2005), 164-172.
[14]	R. Cont and P. Tankov, Financial Modelling With Jump Process, Chapman and Hall/CRC Financial Mathematics Series, Chapman and Hall/CRC, Boca Raton, FL, 2004.
[15]	G. Duvaut and J.-L. Lions, Inequalities In Mechanics and Physics, Springer-Verlag, Berlin, 1976. Translated from the French by C. W. John, Grundlehren der Mathematischen Wissenschaften, 219.
[16]	Y. Fang and W. Chen, A Liouville type theorem for poly-harmonic Dirichlet problems in a half space, Adv. Math., 229 (2012), 2835-2867. doi: 10.1016/j.aim.2012.01.018.
[17]	P. Felmer and A. Quaas, Fundamental solutions and Liouville type properties for nonlinear integral operator, Adv. Math., 226 (2011), 2712-2738. doi: 10.1016/j.aim.2010.09.023.
[18]	M. Moustapha Fall and T. Weth, Monotonicity and nonexistence results for some fractional elliptic problems in the half space,, Available online at \emph{http://arxiv.org/abs/1309.7230}., ().
[19]	M. Moustapha Fall and T. Weth, Nonexistence results for a class of fractional elliptic boundary value problems, J. Funct. Anal., 263 (2012), 2205-2227. doi: 10.1016/j.jfa.2012.06.018.
[20]	R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 1-77. doi: 10.1016/S0370-1573(00)00070-3.
[21]	E. Milakis and L. Silvestre, Regularity for the nonlinear Signorini problem, Adv. Math., 217 (2008), 1301-1312. doi: 10.1016/j.aim.2007.08.009.
[22]	A. Quaas and B. Sirakov, Existence and nonexistence results for fully nonlinear elliptic systems, Indiana Univ. Math. J., 58 (2009), 751-788. doi: 10.1512/iumj.2009.58.3501.
[23]	A. Quaas and A. Xia, Liouville type theorems for nonlinear elliptic equations and systems involving fractional Laplacian in the half space, Calc. Var. Partial Differential Equations, 52 (2015), 641-659. doi: 10.1007/s00526-014-0727-8.
[24]	T. Kulczycki, Properties of Green function of symmetric stable processed, Probability and Mathematical Statistics, 17 (1997), 339-364.
[25]	L. Silvestre, Regularity of the obstacle problem for the fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112. doi: 10.1002/cpa.20153.
[26]	Y. Sire and E. Valdinoci, Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result, J. Funct. Anal., 256 (2009), 1842-1864. doi: 10.1016/j.jfa.2009.01.020.

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References:

[1]	J. Betoin, Lévy Processes, Cambridge Tracts in Mathematics, 121 Cambridge University Press, Cambridge, 1996.
[2]	H. Berestycki and L. Nirenberg, On the method of moving planes and sliding method, Bol. Soc. Brazil. Mat. (N. S.), 22 (1991), 1-37. doi: 10.1007/BF01244896.
[3]	G. Bianchi, Non-existence of positive solutions to semilinear elliptic equations in $R^N$ and $R_{+}^N$ through the method of moving plane, Comm. PDE., 22 (1997), 1671-1690. doi: 10.1080/03605309708821315.
[4]	K. Bogdan, The boundary Harnack principle for the fractional Laplacian, Studia Math., 123 (1997), 43-80.
[5]	X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates, Ann. I. H. Poincaré-AN., 31 (2014), 23-53. doi: 10.1016/j.anihpc.2013.02.001.
[6]	X. Cabré and J. Solà-Morales, Layer solutions in a half-space for boundary reactions, Comm. Pure Appl. Math., 58 (2005), no.12, 1678-1732. doi: 10.1002/cpa.20093.
[7]	L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. in PDE., 32 (2007), 1245-1260. doi: 10.1080/03605300600987306.
[8]	L. Cao and W. Chen, Liouville type theorems for poly-harmonic Navier problems, Disc. Cont. Dyn. Sys., 33 (2013), 3937-3955. doi: 10.3934/dcds.2013.33.3937.
[9]	W. Chen, Y. Fang and R. Yang, Liouville theorems involving the fractional Laplacian on a half space, Advances in Mathematics, 274 (2015), 167-198. doi: 10.1016/j.aim.2014.12.013.
[10]	W. Chen and C. Li, Regularity of solutions for a system of integral equation, Comm. Pure Appl. Anal., 4 (2005), 1-8.
[11]	W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, AIMS. Ser. Differ. Equ. Dyn. Syst., 4 2010.
[12]	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.
[13]	W. Chen, C. Jin, C. Li and J. Lim, Weighted Hardy-Littlewood-Sobolev inequlaities and systems of integral equations, Discrete Contin. Dyn. Syst., (2005), 164-172.
[14]	R. Cont and P. Tankov, Financial Modelling With Jump Process, Chapman and Hall/CRC Financial Mathematics Series, Chapman and Hall/CRC, Boca Raton, FL, 2004.
[15]	G. Duvaut and J.-L. Lions, Inequalities In Mechanics and Physics, Springer-Verlag, Berlin, 1976. Translated from the French by C. W. John, Grundlehren der Mathematischen Wissenschaften, 219.
[16]	Y. Fang and W. Chen, A Liouville type theorem for poly-harmonic Dirichlet problems in a half space, Adv. Math., 229 (2012), 2835-2867. doi: 10.1016/j.aim.2012.01.018.
[17]	P. Felmer and A. Quaas, Fundamental solutions and Liouville type properties for nonlinear integral operator, Adv. Math., 226 (2011), 2712-2738. doi: 10.1016/j.aim.2010.09.023.
[18]	M. Moustapha Fall and T. Weth, Monotonicity and nonexistence results for some fractional elliptic problems in the half space,, Available online at \emph{http://arxiv.org/abs/1309.7230}., ().
[19]	M. Moustapha Fall and T. Weth, Nonexistence results for a class of fractional elliptic boundary value problems, J. Funct. Anal., 263 (2012), 2205-2227. doi: 10.1016/j.jfa.2012.06.018.
[20]	R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 1-77. doi: 10.1016/S0370-1573(00)00070-3.
[21]	E. Milakis and L. Silvestre, Regularity for the nonlinear Signorini problem, Adv. Math., 217 (2008), 1301-1312. doi: 10.1016/j.aim.2007.08.009.
[22]	A. Quaas and B. Sirakov, Existence and nonexistence results for fully nonlinear elliptic systems, Indiana Univ. Math. J., 58 (2009), 751-788. doi: 10.1512/iumj.2009.58.3501.
[23]	A. Quaas and A. Xia, Liouville type theorems for nonlinear elliptic equations and systems involving fractional Laplacian in the half space, Calc. Var. Partial Differential Equations, 52 (2015), 641-659. doi: 10.1007/s00526-014-0727-8.
[24]	T. Kulczycki, Properties of Green function of symmetric stable processed, Probability and Mathematical Statistics, 17 (1997), 339-364.
[25]	L. Silvestre, Regularity of the obstacle problem for the fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112. doi: 10.1002/cpa.20153.
[26]	Y. Sire and E. Valdinoci, Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result, J. Funct. Anal., 256 (2009), 1842-1864. doi: 10.1016/j.jfa.2009.01.020.

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