# American Institute of Mathematical Sciences

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 and Applied Analysis, 2014, 13 (3) : 977-990. doi: 10.3934/cpaa.2014.13.977
##### References:
 [1] H. Berestycki and, L. Nirenberg, On the method of moving planes and the sliding method, Bol. Soc. Brazil. Mat. (N.S.), 22 (1991), 1-37. [2] L. Cao and Z. Dai, A Liouville-type theorem for an integral equations system on a half-space $R^n_+$, Journal of Mathematical Analysis and Applications, 389 (2012), 1365-1373. doi: 10.1016/j.jmaa.2012.01.015. [3] L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math., 42 (1989), 271-297. doi: 10.1002/cpa.3160420304. [4] C. Jin and C. Li, Symmetry of solutions to some integral equations, Proc. Amer. Math. Soc., 134 (2006), 1661-1670. doi: 10.1090/S0002-9939-05-08411-X. [5] W. Chen and C. Li, A priori estimates for prescribing scalar curvature equations, Annals of Math., 145 (1997), 547-564. doi: 10.2307/2951844. [6] W. Chen and C. Li, The best constant in some weighted Hardy-Littlewood-Sobolev inequality, Proc. Amer. Math. Soc., 136 (2008), 955-962. doi: 10.1090/S0002-9939-07-09232-5. [7] W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, AIMS Book Series on Diff. Equa. Dyn. Sys., 4, 2010. [8] W. Chen and C. Li, A sup + inf inequality near $R=0$, Advances in Math., 220 (2009), 219-245. doi: 10.1016/j.aim.2008.09.005. [9] W. Chen and C. Li, An integral system and the Lane-Emden conjecture, Disc. Cont. Dyn. Sys., 4 (2009), 1167-1184. doi: 10.3934/dcds.2009.24.1167. [10] W. Chen, C. Li and Y. Fang, Super-polyharmonic property for a system with Navier conditions on $R^n_+$, submitted to Comm. PDEs, 2012. [11] W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., LLVIII (2005), 1-14. doi: 10.1002/cpa.20116. [12] W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation, Disc. Cont. Dyn. Sys., 12 (2005), 347-354. [13] W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations, Comm. PDEs., 30 (2005), 59-65. doi: 10.1081/PDE-200044445. [14] A. Chang and P. Yang, On uniqueness of an n-th order differential equation in conformal geometry, Math. Res. Letters, 4 (1997), 1-12. [15] L. Fraenkel, An Introduction to Maximum Principles and Symmetry in Elliptic Problems, Cambridge Unversity Press, New York, 2000. doi: 10.1017/CBO9780511569203. [16] Y. Fang and W. Chen, A Liouville type theorem for poly-harmonic Dirichlet problem in a half space, Advances in Math., 229 (2012), 2835-2867. doi: 10.1016/j.aim.2012.01.018. [17] B. Gidas, W. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n$, Mathematical Analysis and Applications, Vol. 7a of the book series Advances in Math., Academic Press, New York, 1981. [18] E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118 (1983), 349-374. doi: 10.2307/2007032. [19] C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Invent. Math., 123 (1996), 221-231. doi: 10.1007/s002220050023. [20] C. Li and L. Ma, Uniqueness of positive bound states to Shrodinger systems with critical exponents, SIAM J. Math. Analysis, 40 (2008), 1049-1057. doi: 10.1137/080712301. [21] C. Liu and S. Qiao, Symmetry and monotonicity for a system of integal equations, Comm. Pure Appl. Anal., 6 (2009), 1925-1932. doi: 10.3934/cpaa.2009.8.1925. [22] D. Li and R. Zhuo, An integral equation on half space, Proc. Amer. Math. Soc., 138 (2010), 2779-2791. doi: 10.1090/S0002-9939-10-10368-2. [23] L. Ma and D. Chen, A Liouville type theorem for an integral system, Comm. Pure Appl. Anal., 5 (2006), 855-859. doi: 10.3934/cpaa.2006.5.855. [24] B. Ou, A remark on a singular integral equation, Houston J. of Math., 25 (1999), 181-184. [25] J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318. [26] R. Zhuo and D. Li, A system of integral equations on half space, Journal of Mathematical Analysis and Applications, 381 (2011), 392-401. doi: 10.1016/j.jmaa.2011.02.060.

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##### References:
 [1] H. Berestycki and, L. Nirenberg, On the method of moving planes and the sliding method, Bol. Soc. Brazil. Mat. (N.S.), 22 (1991), 1-37. [2] L. Cao and Z. Dai, A Liouville-type theorem for an integral equations system on a half-space $R^n_+$, Journal of Mathematical Analysis and Applications, 389 (2012), 1365-1373. doi: 10.1016/j.jmaa.2012.01.015. [3] L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math., 42 (1989), 271-297. doi: 10.1002/cpa.3160420304. [4] C. Jin and C. Li, Symmetry of solutions to some integral equations, Proc. Amer. Math. Soc., 134 (2006), 1661-1670. doi: 10.1090/S0002-9939-05-08411-X. [5] W. Chen and C. Li, A priori estimates for prescribing scalar curvature equations, Annals of Math., 145 (1997), 547-564. doi: 10.2307/2951844. [6] W. Chen and C. Li, The best constant in some weighted Hardy-Littlewood-Sobolev inequality, Proc. Amer. Math. Soc., 136 (2008), 955-962. doi: 10.1090/S0002-9939-07-09232-5. [7] W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, AIMS Book Series on Diff. Equa. Dyn. Sys., 4, 2010. [8] W. Chen and C. Li, A sup + inf inequality near $R=0$, Advances in Math., 220 (2009), 219-245. doi: 10.1016/j.aim.2008.09.005. [9] W. Chen and C. Li, An integral system and the Lane-Emden conjecture, Disc. Cont. Dyn. Sys., 4 (2009), 1167-1184. doi: 10.3934/dcds.2009.24.1167. [10] W. Chen, C. Li and Y. Fang, Super-polyharmonic property for a system with Navier conditions on $R^n_+$, submitted to Comm. PDEs, 2012. [11] W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., LLVIII (2005), 1-14. doi: 10.1002/cpa.20116. [12] W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation, Disc. Cont. Dyn. Sys., 12 (2005), 347-354. [13] W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations, Comm. PDEs., 30 (2005), 59-65. doi: 10.1081/PDE-200044445. [14] A. Chang and P. Yang, On uniqueness of an n-th order differential equation in conformal geometry, Math. Res. Letters, 4 (1997), 1-12. [15] L. Fraenkel, An Introduction to Maximum Principles and Symmetry in Elliptic Problems, Cambridge Unversity Press, New York, 2000. doi: 10.1017/CBO9780511569203. [16] Y. Fang and W. Chen, A Liouville type theorem for poly-harmonic Dirichlet problem in a half space, Advances in Math., 229 (2012), 2835-2867. doi: 10.1016/j.aim.2012.01.018. [17] B. Gidas, W. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n$, Mathematical Analysis and Applications, Vol. 7a of the book series Advances in Math., Academic Press, New York, 1981. [18] E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118 (1983), 349-374. doi: 10.2307/2007032. [19] C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Invent. Math., 123 (1996), 221-231. doi: 10.1007/s002220050023. [20] C. Li and L. Ma, Uniqueness of positive bound states to Shrodinger systems with critical exponents, SIAM J. Math. Analysis, 40 (2008), 1049-1057. doi: 10.1137/080712301. [21] C. Liu and S. Qiao, Symmetry and monotonicity for a system of integal equations, Comm. Pure Appl. Anal., 6 (2009), 1925-1932. doi: 10.3934/cpaa.2009.8.1925. [22] D. Li and R. Zhuo, An integral equation on half space, Proc. Amer. Math. Soc., 138 (2010), 2779-2791. doi: 10.1090/S0002-9939-10-10368-2. [23] L. Ma and D. Chen, A Liouville type theorem for an integral system, Comm. Pure Appl. Anal., 5 (2006), 855-859. doi: 10.3934/cpaa.2006.5.855. [24] B. Ou, A remark on a singular integral equation, Houston J. of Math., 25 (1999), 181-184. [25] J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318. [26] R. Zhuo and D. Li, A system of integral equations on half space, Journal of Mathematical Analysis and Applications, 381 (2011), 392-401. doi: 10.1016/j.jmaa.2011.02.060.
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