    May  2014, 13(3): 977-990. doi: 10.3934/cpaa.2014.13.977

## Liouville type theorems for Schrödinger system with Navier boundary conditions in a half space

 1 Department of Mathematical Sciences, Yeshiva University, New York, NY 10033, United States 2 School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024 3 College of Mathematics and Information Science, Nanchang Hangkong University, Nanchang, Jiangxi 330063, China

Received  November 2012 Revised  August 2013 Published  December 2013

In this paper, we study the positive solutions for the following integral system: \begin{eqnarray} u(x)=\int_{R^n_+}(\frac{1}{|x-y|^{n-\alpha}}-\frac{1}{|x^*-y|^{n-\alpha}})u^{\beta_1}(y)v^{\gamma_1}(y)dy ,\\ v(x)=\int_{R^n_+}(\frac{1}{|x-y|^{n-\alpha}}-\frac{1}{|x^*-y|^{n-\alpha}})u^{\beta_2}(y)v^{\gamma_2}(y)dy, \end{eqnarray} where $0 < \alpha < n$ and $x^*=(x_1,\cdots,x_{n-1},-x_n)$ is the reflection of the point $x$ about the plane $R^{n-1}$, and $\beta_1, \gamma_1, \beta_2, \gamma_2$ satisfy the condition$(f_1)$: \begin{eqnarray} 1 \leq \beta_1,\gamma_1,\beta_2,\gamma_2 \leq \frac{n+\alpha}{n-\alpha}\ \mbox{with}\ \beta_1+\gamma_1= \frac{n+\alpha}{n-\alpha}=\beta_2+\gamma_2, \beta_1\neq \beta_2, \gamma_1 \neq \gamma_2. \end{eqnarray}

This integral system is closely related to the PDE system with Navier boundary conditions, when $\alpha$ is a even number between $0$ and $n$, \begin{eqnarray} (- \Delta)^{\frac{\alpha}{2}}u(x)=u^{\beta_1}(x)v^{\gamma_1}(x), \mbox{in}\ R^n_+,\\ (- \Delta)^{\frac{\alpha}{2}}v(x)=u^{\beta_2}(x)v^{\gamma_2}(x), \mbox{in}\ R^n_+,\\ u(x)=-\Delta u(x)=\cdots =(-\Delta)^{\frac{\alpha}{2}-1} u(x)=0,\mbox{on}\ \partial{R^n_+},\\ v(x)=-\Delta v(x)=\cdots =(-\Delta)^{\frac{\alpha}{2}-1} v(x)=0,\mbox{on}\ \partial{R^n_+}. \end{eqnarray}

More precisely, any solution of (1) multiplied by a constant satisfies (2). We use method of moving planes in integral forms introduced by Chen-Li-Ou to derive rotational symmetry, monotonicity, and non-existence of the positive solutions of (1) on the half space $R^n_+$.
Citation: 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
##### References:
  H. Berestycki and, L. Nirenberg, On the method of moving planes and the sliding method,, \emph{Bol. Soc. Brazil. Mat. (N.S.)}, 22 (1991), 1.   Google Scholar  L. Cao and Z. Dai, A Liouville-type theorem for an integral equations system on a half-space $R^n_+$,, \emph{Journal of Mathematical Analysis and Applications}, 389 (2012), 1365.  doi: 10.1016/j.jmaa.2012.01.015.  Google Scholar  L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth,, \emph{Comm. Pure Appl. Math.}, 42 (1989), 271.  doi: 10.1002/cpa.3160420304.  Google Scholar  C. Jin and C. Li, Symmetry of solutions to some integral equations,, \emph{Proc. Amer. Math. Soc.}, 134 (2006), 1661.  doi: 10.1090/S0002-9939-05-08411-X.  Google Scholar  W. Chen and C. Li, A priori estimates for prescribing scalar curvature equations,, \emph{Annals of Math.}, 145 (1997), 547.  doi: 10.2307/2951844.  Google Scholar  W. Chen and C. Li, The best constant in some weighted Hardy-Littlewood-Sobolev inequality,, \emph{Proc. Amer. Math. Soc.}, 136 (2008), 955.  doi: 10.1090/S0002-9939-07-09232-5.  Google Scholar  W. Chen and C. Li, Methods on Nonlinear Elliptic Equations,, AIMS Book Series on Diff. Equa. Dyn. Sys., 4 (2010). Google Scholar  W. Chen and C. Li, A sup + inf inequality near $R=0$,, \emph{Advances in Math.}, 220 (2009), 219.  doi: 10.1016/j.aim.2008.09.005.  Google Scholar  W. Chen and C. Li, An integral system and the Lane-Emden conjecture,, \emph{Disc. Cont. Dyn. Sys.}, 4 (2009), 1167.  doi: 10.3934/dcds.2009.24.1167.  Google Scholar  W. Chen, C. Li and Y. Fang, Super-polyharmonic property for a system with Navier conditions on $R^n_+$,, submitted to Comm. PDEs, (2012).   Google Scholar  W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation,, \emph{Comm. Pure Appl. Math.}, LLVIII (2005), 1.  doi: 10.1002/cpa.20116.  Google Scholar  W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation,, \emph{Disc. Cont. Dyn. Sys.}, 12 (2005), 347. Google Scholar  W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations,, \emph{Comm. PDEs.}, 30 (2005), 59.  doi: 10.1081/PDE-200044445.  Google Scholar  A. Chang and P. Yang, On uniqueness of an n-th order differential equation in conformal geometry,, \emph{Math. Res. Letters}, 4 (1997), 1.   Google Scholar  L. Fraenkel, An Introduction to Maximum Principles and Symmetry in Elliptic Problems,, Cambridge Unversity Press, (2000).  doi: 10.1017/CBO9780511569203.  Google Scholar  Y. Fang and W. Chen, A Liouville type theorem for poly-harmonic Dirichlet problem in a half space,, \emph{Advances in Math.}, 229 (2012), 2835.  doi: 10.1016/j.aim.2012.01.018.  Google Scholar  B. Gidas, W. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n$,, Mathematical Analysis and Applications, (1981). Google Scholar  E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities,, \emph{Ann. of Math.}, 118 (1983), 349.  doi: 10.2307/2007032.  Google Scholar  C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations,, \emph{Invent. Math.}, 123 (1996), 221.  doi: 10.1007/s002220050023.  Google Scholar  C. Li and L. Ma, Uniqueness of positive bound states to Shrodinger systems with critical exponents,, \emph{SIAM J. Math. Analysis}, 40 (2008), 1049.  doi: 10.1137/080712301.  Google Scholar  C. Liu and S. Qiao, Symmetry and monotonicity for a system of integal equations,, \emph{Comm. Pure Appl. Anal.}, 6 (2009), 1925.  doi: 10.3934/cpaa.2009.8.1925.  Google Scholar  D. Li and R. Zhuo, An integral equation on half space,, \emph{Proc. Amer. Math. Soc.}, 138 (2010), 2779.  doi: 10.1090/S0002-9939-10-10368-2.  Google Scholar  L. Ma and D. Chen, A Liouville type theorem for an integral system,, \emph{Comm. Pure Appl. Anal.}, 5 (2006), 855.  doi: 10.3934/cpaa.2006.5.855.  Google Scholar  B. Ou, A remark on a singular integral equation,, \emph{Houston J. of Math.}, 25 (1999), 181. Google Scholar  J. Serrin, A symmetry problem in potential theory,, \emph{Arch. Rational Mech. Anal.}, 43 (1971), 304. Google Scholar  R. Zhuo and D. Li, A system of integral equations on half space,, \emph{Journal of Mathematical Analysis and Applications}, 381 (2011), 392.  doi: 10.1016/j.jmaa.2011.02.060.  Google Scholar

show all references

##### References:
  H. Berestycki and, L. Nirenberg, On the method of moving planes and the sliding method,, \emph{Bol. Soc. Brazil. Mat. (N.S.)}, 22 (1991), 1.   Google Scholar  L. Cao and Z. Dai, A Liouville-type theorem for an integral equations system on a half-space $R^n_+$,, \emph{Journal of Mathematical Analysis and Applications}, 389 (2012), 1365.  doi: 10.1016/j.jmaa.2012.01.015.  Google Scholar  L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth,, \emph{Comm. Pure Appl. Math.}, 42 (1989), 271.  doi: 10.1002/cpa.3160420304.  Google Scholar  C. Jin and C. Li, Symmetry of solutions to some integral equations,, \emph{Proc. Amer. Math. Soc.}, 134 (2006), 1661.  doi: 10.1090/S0002-9939-05-08411-X.  Google Scholar  W. Chen and C. Li, A priori estimates for prescribing scalar curvature equations,, \emph{Annals of Math.}, 145 (1997), 547.  doi: 10.2307/2951844.  Google Scholar  W. Chen and C. Li, The best constant in some weighted Hardy-Littlewood-Sobolev inequality,, \emph{Proc. Amer. Math. Soc.}, 136 (2008), 955.  doi: 10.1090/S0002-9939-07-09232-5.  Google Scholar  W. Chen and C. Li, Methods on Nonlinear Elliptic Equations,, AIMS Book Series on Diff. Equa. Dyn. Sys., 4 (2010). Google Scholar  W. Chen and C. Li, A sup + inf inequality near $R=0$,, \emph{Advances in Math.}, 220 (2009), 219.  doi: 10.1016/j.aim.2008.09.005.  Google Scholar  W. Chen and C. Li, An integral system and the Lane-Emden conjecture,, \emph{Disc. Cont. Dyn. Sys.}, 4 (2009), 1167.  doi: 10.3934/dcds.2009.24.1167.  Google Scholar  W. Chen, C. Li and Y. Fang, Super-polyharmonic property for a system with Navier conditions on $R^n_+$,, submitted to Comm. PDEs, (2012).   Google Scholar  W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation,, \emph{Comm. Pure Appl. Math.}, LLVIII (2005), 1.  doi: 10.1002/cpa.20116.  Google Scholar  W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation,, \emph{Disc. Cont. Dyn. Sys.}, 12 (2005), 347. Google Scholar  W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations,, \emph{Comm. PDEs.}, 30 (2005), 59.  doi: 10.1081/PDE-200044445.  Google Scholar  A. Chang and P. Yang, On uniqueness of an n-th order differential equation in conformal geometry,, \emph{Math. Res. Letters}, 4 (1997), 1.   Google Scholar  L. Fraenkel, An Introduction to Maximum Principles and Symmetry in Elliptic Problems,, Cambridge Unversity Press, (2000).  doi: 10.1017/CBO9780511569203.  Google Scholar  Y. Fang and W. Chen, A Liouville type theorem for poly-harmonic Dirichlet problem in a half space,, \emph{Advances in Math.}, 229 (2012), 2835.  doi: 10.1016/j.aim.2012.01.018.  Google Scholar  B. Gidas, W. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n$,, Mathematical Analysis and Applications, (1981). Google Scholar  E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities,, \emph{Ann. of Math.}, 118 (1983), 349.  doi: 10.2307/2007032.  Google Scholar  C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations,, \emph{Invent. Math.}, 123 (1996), 221.  doi: 10.1007/s002220050023.  Google Scholar  C. Li and L. Ma, Uniqueness of positive bound states to Shrodinger systems with critical exponents,, \emph{SIAM J. Math. Analysis}, 40 (2008), 1049.  doi: 10.1137/080712301.  Google Scholar  C. Liu and S. Qiao, Symmetry and monotonicity for a system of integal equations,, \emph{Comm. Pure Appl. Anal.}, 6 (2009), 1925.  doi: 10.3934/cpaa.2009.8.1925.  Google Scholar  D. Li and R. Zhuo, An integral equation on half space,, \emph{Proc. Amer. Math. Soc.}, 138 (2010), 2779.  doi: 10.1090/S0002-9939-10-10368-2.  Google Scholar  L. Ma and D. Chen, A Liouville type theorem for an integral system,, \emph{Comm. Pure Appl. Anal.}, 5 (2006), 855.  doi: 10.3934/cpaa.2006.5.855.  Google Scholar  B. Ou, A remark on a singular integral equation,, \emph{Houston J. of Math.}, 25 (1999), 181. Google Scholar  J. Serrin, A symmetry problem in potential theory,, \emph{Arch. Rational Mech. Anal.}, 43 (1971), 304. Google Scholar  R. Zhuo and D. Li, A system of integral equations on half space,, \emph{Journal of Mathematical Analysis and Applications}, 381 (2011), 392.  doi: 10.1016/j.jmaa.2011.02.060.  Google Scholar
  Changlu Liu, Shuangli Qiao. Symmetry and monotonicity for a system of integral equations. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1925-1932. doi: 10.3934/cpaa.2009.8.1925  Yingshu Lü. Symmetry and non-existence of solutions to an integral system. Communications on Pure & Applied Analysis, 2018, 17 (3) : 807-821. doi: 10.3934/cpaa.2018041  Yingshu Lü, Chunqin Zhou. Symmetry for an integral system with general nonlinearity. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1533-1543. doi: 10.3934/dcds.2018121  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  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  Abdelkader Boucherif. Positive Solutions of second order differential equations with integral boundary conditions. Conference Publications, 2007, 2007 (Special) : 155-159. doi: 10.3934/proc.2007.2007.155  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  Patricia J.Y. Wong. Existence of solutions to singular integral equations. Conference Publications, 2009, 2009 (Special) : 818-827. doi: 10.3934/proc.2009.2009.818  Dorina Mitrea and Marius Mitrea. Boundary integral methods for harmonic differential forms in Lipschitz domains. Electronic Research Announcements, 1996, 2: 92-97.  Wenxiong Chen, Congming Li. Regularity of solutions for a system of integral equations. Communications on Pure & Applied Analysis, 2005, 4 (1) : 1-8. doi: 10.3934/cpaa.2005.4.1  Johnny Henderson, Rodica Luca. Existence of positive solutions for a system of nonlinear second-order integral boundary value problems. Conference Publications, 2015, 2015 (special) : 596-604. doi: 10.3934/proc.2015.0596  Xiaoxue Ji, Pengcheng Niu, Pengyan Wang. Non-existence results for cooperative semi-linear fractional system via direct method of moving spheres. Communications on Pure & Applied Analysis, 2020, 19 (2) : 1111-1128. doi: 10.3934/cpaa.2020051  Gennaro Infante. Eigenvalues and positive solutions of odes involving integral boundary conditions. Conference Publications, 2005, 2005 (Special) : 436-442. doi: 10.3934/proc.2005.2005.436  Carlos Fresneda-Portillo, Sergey E. Mikhailov. Analysis of Boundary-Domain Integral Equations to the mixed BVP for a compressible stokes system with variable viscosity. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3059-3088. doi: 10.3934/cpaa.2019137  Olusola Kolebaje, Ebenezer Bonyah, Lateef Mustapha. The first integral method for two fractional non-linear biological models. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 487-502. doi: 10.3934/dcdss.2019032  Thomas Y. Hou, Pingwen Zhang. Convergence of a boundary integral method for 3-D water waves. Discrete & Continuous Dynamical Systems - B, 2002, 2 (1) : 1-34. doi: 10.3934/dcdsb.2002.2.1  Wu Chen, Zhongxue Lu. Existence and nonexistence of positive solutions to an integral system involving Wolff potential. Communications on Pure & Applied Analysis, 2016, 15 (2) : 385-398. doi: 10.3934/cpaa.2016.15.385  Z. K. Eshkuvatov, M. Kammuji, Bachok M. Taib, N. M. A. Nik Long. Effective approximation method for solving linear Fredholm-Volterra integral equations. Numerical Algebra, Control & Optimization, 2017, 7 (1) : 77-88. doi: 10.3934/naco.2017004  Stanisław Migórski, Shengda Zeng. The Rothe method for multi-term time fractional integral diffusion equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 719-735. doi: 10.3934/dcdsb.2018204  Franck Boyer, Pierre Fabrie. Outflow boundary conditions for the incompressible non-homogeneous Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2007, 7 (2) : 219-250. doi: 10.3934/dcdsb.2007.7.219

2018 Impact Factor: 0.925