American Institute of Mathematical Sciences

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

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

Received  March 2015 Revised  July 2015 Published  September 2015

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.
Citation: De Tang, Yanqin Fang. Regularity and nonexistence of solutions for a system involving the fractional Laplacian. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2431-2451. doi: 10.3934/cpaa.2015.14.2431
References:
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References:
 [1] J. Betoin, Lévy Processes, Cambridge Tracts in Mathematics, 121 Cambridge University Press, Cambridge, 1996.  Google Scholar [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.  Google Scholar [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.  Google Scholar [4] K. Bogdan, The boundary Harnack principle for the fractional Laplacian, Studia Math., 123 (1997), 43-80.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [10] W. Chen and C. Li, Regularity of solutions for a system of integral equation, Comm. Pure Appl. Anal., 4 (2005), 1-8.  Google Scholar [11] W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, AIMS. Ser. Differ. Equ. Dyn. Syst., 4 2010.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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}., ().   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [24] T. Kulczycki, Properties of Green function of symmetric stable processed, Probability and Mathematical Statistics, 17 (1997), 339-364.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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