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:
[1]

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

[2]

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

[3]

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

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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

[5]

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

[6]

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

[7]

W. Chen and C. Li, Methods on Nonlinear Elliptic Equations,, AIMS Book Series on Diff. Equa. Dyn. Sys., 4 (2010). Google Scholar

[8]

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

[9]

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

[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). Google Scholar

[11]

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

[12]

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

[13]

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

[14]

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

[15]

L. Fraenkel, An Introduction to Maximum Principles and Symmetry in Elliptic Problems,, Cambridge Unversity Press, (2000). doi: 10.1017/CBO9780511569203. Google Scholar

[16]

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

[17]

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

[18]

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

[19]

C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations,, \emph{Invent. Math.}, 123 (1996), 221. doi: 10.1007/s002220050023. Google Scholar

[20]

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

[21]

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

[22]

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

[23]

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

[24]

B. Ou, A remark on a singular integral equation,, \emph{Houston J. of Math.}, 25 (1999), 181. Google Scholar

[25]

J. Serrin, A symmetry problem in potential theory,, \emph{Arch. Rational Mech. Anal.}, 43 (1971), 304. Google Scholar

[26]

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:
[1]

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

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

W. Chen and C. Li, Methods on Nonlinear Elliptic Equations,, AIMS Book Series on Diff. Equa. Dyn. Sys., 4 (2010). Google Scholar

[8]

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

[9]

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

[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). Google Scholar

[11]

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

[12]

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

[13]

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

[14]

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

[15]

L. Fraenkel, An Introduction to Maximum Principles and Symmetry in Elliptic Problems,, Cambridge Unversity Press, (2000). doi: 10.1017/CBO9780511569203. Google Scholar

[16]

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

[17]

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

[18]

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

[19]

C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations,, \emph{Invent. Math.}, 123 (1996), 221. doi: 10.1007/s002220050023. Google Scholar

[20]

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

[21]

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

[22]

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

[23]

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

[24]

B. Ou, A remark on a singular integral equation,, \emph{Houston J. of Math.}, 25 (1999), 181. Google Scholar

[25]

J. Serrin, A symmetry problem in potential theory,, \emph{Arch. Rational Mech. Anal.}, 43 (1971), 304. Google Scholar

[26]

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

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