Article Contents
Article Contents

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

• 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 $R^n_+$.
Mathematics Subject Classification: Primary: 31A10, 35B45; Secondary: 35B53, 35J91.

 Citation:

•  [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.