-
Previous Article
Consequences of the choice of a particular basis of $L^2(S^3)$ for the cubic wave equation on the sphere and the Euclidean space
- CPAA Home
- This Issue
- Next Article
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 |
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 $R^n_+$.
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. |
show all references
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. |
[1] |
Changlu Liu, Shuangli Qiao. Symmetry and monotonicity for a system of integral equations. Communications on Pure and Applied Analysis, 2009, 8 (6) : 1925-1932. doi: 10.3934/cpaa.2009.8.1925 |
[2] |
Yingshu Lü. Symmetry and non-existence of solutions to an integral system. Communications on Pure and Applied Analysis, 2018, 17 (3) : 807-821. doi: 10.3934/cpaa.2018041 |
[3] |
Yingshu Lü, Chunqin Zhou. Symmetry for an integral system with general nonlinearity. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1533-1543. doi: 10.3934/dcds.2018121 |
[4] |
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 |
[5] |
Baiyu Liu. Direct method of moving planes for logarithmic Laplacian system in bounded domains. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 5339-5349. doi: 10.3934/dcds.2018235 |
[6] |
Pengyan Wang, Pengcheng Niu. A direct method of moving planes for a fully nonlinear nonlocal system. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1707-1718. doi: 10.3934/cpaa.2017082 |
[7] |
Meixia Dou. A direct method of moving planes for fractional Laplacian equations in the unit ball. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1797-1807. doi: 10.3934/cpaa.2016015 |
[8] |
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 |
[9] |
Dorina Mitrea and Marius Mitrea. Boundary integral methods for harmonic differential forms in Lipschitz domains. Electronic Research Announcements, 1996, 2: 92-97. |
[10] |
Wenxiong Chen, Congming Li. Regularity of solutions for a system of integral equations. Communications on Pure and Applied Analysis, 2005, 4 (1) : 1-8. doi: 10.3934/cpaa.2005.4.1 |
[11] |
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 |
[12] |
Xiaoxue Ji, Pengcheng Niu, Pengyan Wang. Non-existence results for cooperative semi-linear fractional system via direct method of moving spheres. Communications on Pure and Applied Analysis, 2020, 19 (2) : 1111-1128. doi: 10.3934/cpaa.2020051 |
[13] |
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 |
[14] |
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 and Applied Analysis, 2019, 18 (6) : 3059-3088. doi: 10.3934/cpaa.2019137 |
[15] |
Olusola Kolebaje, Ebenezer Bonyah, Lateef Mustapha. The first integral method for two fractional non-linear biological models. Discrete and Continuous Dynamical Systems - S, 2019, 12 (3) : 487-502. doi: 10.3934/dcdss.2019032 |
[16] |
Thomas Y. Hou, Pingwen Zhang. Convergence of a boundary integral method for 3-D water waves. Discrete and Continuous Dynamical Systems - B, 2002, 2 (1) : 1-34. doi: 10.3934/dcdsb.2002.2.1 |
[17] |
Nguyen Anh Tuan, Donal O'Regan, Dumitru Baleanu, Nguyen H. Tuan. On time fractional pseudo-parabolic equations with nonlocal integral conditions. Evolution Equations and Control Theory, 2022, 11 (1) : 225-238. doi: 10.3934/eect.2020109 |
[18] |
Wu Chen, Zhongxue Lu. Existence and nonexistence of positive solutions to an integral system involving Wolff potential. Communications on Pure and Applied Analysis, 2016, 15 (2) : 385-398. doi: 10.3934/cpaa.2016.15.385 |
[19] |
Yuxia Guo, Shaolong Peng. A direct method of moving planes for fully nonlinear nonlocal operators and applications. Discrete and Continuous Dynamical Systems - S, 2021, 14 (6) : 1871-1897. doi: 10.3934/dcdss.2020462 |
[20] |
Stanisław Migórski, Shengda Zeng. The Rothe method for multi-term time fractional integral diffusion equations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 719-735. doi: 10.3934/dcdsb.2018204 |
2020 Impact Factor: 1.916
Tools
Metrics
Other articles
by authors
[Back to Top]