September  2013, 33(9): 3937-3955. doi: 10.3934/dcds.2013.33.3937

Liouville type theorems for poly-harmonic Navier problems

1. 

College of Mathematics and Information Science, Henan Normal University, Henan, 453007, China

2. 

Department of Mathematics, Yeshiva University, New York, NY 10033

Received  May 2012 Revised  December 2012 Published  March 2013

In this paper we consider the following semi-linear poly-harmonic equation with Navier boundary conditions on the half space $R^n_+$: \begin{equation} \left\{\begin{array}{l} (-\triangle)^{\frac{\alpha}{2}} u=u^p,\ \ \ \ \ \:\:\: \:\:\:\:\:\ \:\:\ \ \ \ \ \ \ \ \ \ \ \ \:\:\:\:\ \mbox{in}\,\ R^n_+,\\ u=-\triangle u=\cdots=(-\triangle)^{\frac{\alpha}{2}-1}u=0, \ \ \ \mbox{on}\ \partial R^n_+, \end{array} \right. \label{phe1} \end{equation} where $\alpha$ is any even number between $0$ and $n$, and $p>1$.
    First we prove that (1) is equivalent to the following integral equation \begin{equation} u(x)=\int_{R^n_+}G(x,y,\alpha) u^p(y)dy,\,\,\,\,\, x\in\,R^n_+, \label{ie0} \end{equation} under some very mild growth condition, where $G(x, y,\alpha)$ is the Green's function associated with the same Navier boundary conditions on the half-space .
    Then by combining the method of moving planes in integral forms with a certain type of Kelvin transform, we obtain the non-existence of positive solutions for integral equation (2) in both subcritical and critical cases under only local integrability conditions. This remarkably weaken the global integrability assumptions on solutions in paper [3]. Our results on integral equation (2) are valid for all real values $\alpha$ between $0$ and $n$.
    Finally, we establish a Liouville type theorem for PDE (1), and this generalizes Guo and Liu's result [21] by significantly weaken the growth conditions on the solutions.
Citation: Linfen Cao, Wenxiong Chen. Liouville type theorems for poly-harmonic Navier problems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 3937-3955. doi: 10.3934/dcds.2013.33.3937
References:
[1]

G. Bianchi, Non-existence of positive solutions to semilinear elliptic equations in $R^N$ and $R_+^N$ through the method of moving planes,, Comm. PDE., 22 (1997), 1671. doi: 10.1080/03605309708821315. Google Scholar

[2]

H. Berestycki and L. Nirenberg, On the method of moving planes and sliding method,, Bol. Soc. Brazil. Mat. (N. S.), 22 (1991), 1. doi: 10.1007/BF01244896. Google Scholar

[3]

L. Cao and Z. Dai, A Liouville-type theorem for an integral equation on a half-space $R^n_+$,, J. Math. Anal. Appl., 389 (2012), 1365. doi: 10.1016/j.jmaa.2012.01.015. Google Scholar

[4]

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

[5]

W. Chen and C. Li, Classification of positive solutions for nonlinear differential and integral systems with critical exponents,, Acta Math. Sci. Ser. B Engl. Ed., 29 (2009), 949. doi: 10.1016/S0252-9602(09)60079-5. Google Scholar

[6]

W. Chen and C. Li, An integral system and the Lane-Emden conjecture,, Disc. Cont. Dyn. Sys., 24 (2009), 1167. doi: 10.3934/dcds.2009.24.1167. Google Scholar

[7]

W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation,, Comn. Pure Appl. Math., 59 (2006), 330. doi: 10.1002/cpa.20116. Google Scholar

[8]

W. Chen, C. Jin, C. Li and J. Lim, Weighted Hardy-Littlewood-Sobolev inequalities and system of integral equations,, Disc. Cont. Dyn. Sys., 2005 (2005), 164. Google Scholar

[9]

W. Chen, C. Li and B. Ou, Qualitative problems of solutions for an integral equation,, Disc. Cont. Dyn. Sys., 12 (2005), 347. Google Scholar

[10]

W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations,, Comm. PDEs, 30 (2005), 59. doi: 10.1081/PDE-200044445. Google Scholar

[11]

W. Chen and C. Li, Super polyharmonic property of solutions for PDE systems and its applications,, to appear in Comm. Pure Appl. Anal., (2012). Google Scholar

[12]

W. Chen and C. Li, Moving planes, moving spheres, and a priori estimates,, J. Diff. Equ., 195 (2003), 1. doi: 10.1016/j.jde.2003.06.004. Google Scholar

[13]

W. Chen and C. Li, A priori estimates for prescribing scalar curvature equations,, Ann. Math. (2), 145 (1997), 547. doi: 10.2307/2951844. Google Scholar

[14]

W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations,, Duke Math. J., 63 (1991), 615. doi: 10.1215/S0012-7094-91-06325-8. Google Scholar

[15]

W. Chen and C. Li, A sup + inf inequality near $R=0$,, Adv. in Math., 220 (2009), 219. doi: 10.1016/j.aim.2008.09.005. Google Scholar

[16]

Super polyharmonic property of solutions of Navier boundary problem in $R^n_+$, preprint,, (2012)., (2012). Google Scholar

[17]

S.-Y. A. Chang and P. Yang, On uniqueness of an nth order differential equation in conformal geometry,, Math. Res. Letters, 4 (1997), 91. Google Scholar

[18]

W. Chen and J. Zhu, Radial symmetry and regularity of solutions for poly-harmonic Drichlet problems,, J. Math. Anal. Appl., 377 (2011), 744. doi: 10.1016/j.jmaa.2010.11.035. Google Scholar

[19]

Y. Fang and W. Chen, A Liouville type theorem for poly-harmonic Dirichlet problems in a half space,, Adv. Math., 229 (2012), 2835. doi: 10.1016/j.aim.2012.01.018. Google Scholar

[20]

Y. Fang and J. Zhang, Nonexistence of positive solution for an integral equation on a half-space $R_+^n$,, Comm. Pure Appl. Anal., 12 (2013), 663. Google Scholar

[21]

Y. Guo and J. Liu, Liouville-type theorems for polyharmonic equations in $R^N$ and in $R_+^N$,, Proc. R. Soc. Edinb. Sect. A, 138 (2008), 339. doi: 10.1017/S0308210506000394. Google Scholar

[22]

B. Gidas, W. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n$,, in, 7a (1981). Google Scholar

[23]

B. Gidas and J. Spruck, A priori bounds for positive solutiions of nonlinear elliptic equations,, Comm. PDEs, 6 (1981), 883. doi: 10.1080/03605308108820196. Google Scholar

[24]

F. Hang, On the integral systems related to Hardy-Littlewood-Sobolev inequality,, Math. Res. Lett., 14 (2007), 373. Google Scholar

[25]

C. Jin and C. Li, Symmetry of solutions to some systems of integral equations,, Proc. AMS, 134 (2006), 1661. doi: 10.1090/S0002-9939-05-08411-X. Google Scholar

[26]

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

[27]

D. Li, G. Ströhmer and L. Wang, Symmetry of integral equations on bounded domains,, Proc. AMS, 137 (2009), 3695. doi: 10.1090/S0002-9939-09-09987-0. Google Scholar

[28]

Y. Y. Li, Remarks on some conformally invariant integral equations: The method of moving spheres,, J. Euro. Math. Soc., 6 (2004), 153. Google Scholar

[29]

C. Li and L. Ma, Uniqueness of positive bound states to Shrodinger systems with critical exponents,, SIAM J. of Appl. Anal., 40 (2008), 1049. doi: 10.1137/080712301. Google Scholar

[30]

C. Liu and S. Qiao, Symmetry and monotonicity for a system of integral equations,, Comm. Pure Appl. Anal., 8 (2009), 1925. doi: 10.3934/cpaa.2009.8.1925. Google Scholar

[31]

C.-S. Lin, A classification of solutions of a conformally invariant fourth order equation in $R^n$,, Comment. Math. Helv., 73 (1998), 206. doi: 10.1007/s000140050052. Google Scholar

[32]

S. Liu, Regularity, symmetry, and uniqueness of some integral type quasilinear equations,, Nonl. Anal., 71 (2009), 1796. doi: 10.1016/j.na.2009.01.014. Google Scholar

[33]

Y. Li and M. Zhu, Uniqueness theorems through the method of moving spheres,, Duke math. J., 80 (1995), 383. doi: 10.1215/S0012-7094-95-08016-8. Google Scholar

[34]

D. Li and R. Zhuo, An integral equation on half space,, Proc. AMS, 138 (2010), 2779. doi: 10.1090/S0002-9939-10-10368-2. Google Scholar

[35]

G. Lu and J. Zhu, Axial symmetry and regularity of solutions to an integral equation in a half space,, Pacific J. Math., 253 (2011), 455. doi: 10.2140/pjm.2011.253.455. Google Scholar

[36]

C. Ma, W. Chen and C. Li, Regularity of solutions for an integral system of Wolff type,, Adv. Math., 226 (2011), 2676. doi: 10.1016/j.aim.2010.07.020. Google Scholar

[37]

L. Ma and D. Chen, A Liouville type theorem for an integral system,, Commun. Pure Appl. Anal., 5 (2006), 855. doi: 10.3934/cpaa.2006.5.855. Google Scholar

[38]

L. Ma and B. Liu, Symmetry results for decay solutions of elliptic systems in the whole space,, Adv. Math., 225 (2010), 3052. doi: 10.1016/j.aim.2010.05.022. Google Scholar

[39]

L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation,, Arch. Rat. Mech. Anal., 195 (2010), 455. doi: 10.1007/s00205-008-0208-3. Google Scholar

[40]

W. Reichel and T. Weth, A priori bounds and a Liouville theorem on a half-space for higher-order elliptic Dirichlet problems,, Math. Z., 261 (2009), 805. doi: 10.1007/s00209-008-0352-3. Google Scholar

[41]

J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations,, Math. Ann., 313 (1999), 207. doi: 10.1007/s002080050258. Google Scholar

show all references

References:
[1]

G. Bianchi, Non-existence of positive solutions to semilinear elliptic equations in $R^N$ and $R_+^N$ through the method of moving planes,, Comm. PDE., 22 (1997), 1671. doi: 10.1080/03605309708821315. Google Scholar

[2]

H. Berestycki and L. Nirenberg, On the method of moving planes and sliding method,, Bol. Soc. Brazil. Mat. (N. S.), 22 (1991), 1. doi: 10.1007/BF01244896. Google Scholar

[3]

L. Cao and Z. Dai, A Liouville-type theorem for an integral equation on a half-space $R^n_+$,, J. Math. Anal. Appl., 389 (2012), 1365. doi: 10.1016/j.jmaa.2012.01.015. Google Scholar

[4]

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

[5]

W. Chen and C. Li, Classification of positive solutions for nonlinear differential and integral systems with critical exponents,, Acta Math. Sci. Ser. B Engl. Ed., 29 (2009), 949. doi: 10.1016/S0252-9602(09)60079-5. Google Scholar

[6]

W. Chen and C. Li, An integral system and the Lane-Emden conjecture,, Disc. Cont. Dyn. Sys., 24 (2009), 1167. doi: 10.3934/dcds.2009.24.1167. Google Scholar

[7]

W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation,, Comn. Pure Appl. Math., 59 (2006), 330. doi: 10.1002/cpa.20116. Google Scholar

[8]

W. Chen, C. Jin, C. Li and J. Lim, Weighted Hardy-Littlewood-Sobolev inequalities and system of integral equations,, Disc. Cont. Dyn. Sys., 2005 (2005), 164. Google Scholar

[9]

W. Chen, C. Li and B. Ou, Qualitative problems of solutions for an integral equation,, Disc. Cont. Dyn. Sys., 12 (2005), 347. Google Scholar

[10]

W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations,, Comm. PDEs, 30 (2005), 59. doi: 10.1081/PDE-200044445. Google Scholar

[11]

W. Chen and C. Li, Super polyharmonic property of solutions for PDE systems and its applications,, to appear in Comm. Pure Appl. Anal., (2012). Google Scholar

[12]

W. Chen and C. Li, Moving planes, moving spheres, and a priori estimates,, J. Diff. Equ., 195 (2003), 1. doi: 10.1016/j.jde.2003.06.004. Google Scholar

[13]

W. Chen and C. Li, A priori estimates for prescribing scalar curvature equations,, Ann. Math. (2), 145 (1997), 547. doi: 10.2307/2951844. Google Scholar

[14]

W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations,, Duke Math. J., 63 (1991), 615. doi: 10.1215/S0012-7094-91-06325-8. Google Scholar

[15]

W. Chen and C. Li, A sup + inf inequality near $R=0$,, Adv. in Math., 220 (2009), 219. doi: 10.1016/j.aim.2008.09.005. Google Scholar

[16]

Super polyharmonic property of solutions of Navier boundary problem in $R^n_+$, preprint,, (2012)., (2012). Google Scholar

[17]

S.-Y. A. Chang and P. Yang, On uniqueness of an nth order differential equation in conformal geometry,, Math. Res. Letters, 4 (1997), 91. Google Scholar

[18]

W. Chen and J. Zhu, Radial symmetry and regularity of solutions for poly-harmonic Drichlet problems,, J. Math. Anal. Appl., 377 (2011), 744. doi: 10.1016/j.jmaa.2010.11.035. Google Scholar

[19]

Y. Fang and W. Chen, A Liouville type theorem for poly-harmonic Dirichlet problems in a half space,, Adv. Math., 229 (2012), 2835. doi: 10.1016/j.aim.2012.01.018. Google Scholar

[20]

Y. Fang and J. Zhang, Nonexistence of positive solution for an integral equation on a half-space $R_+^n$,, Comm. Pure Appl. Anal., 12 (2013), 663. Google Scholar

[21]

Y. Guo and J. Liu, Liouville-type theorems for polyharmonic equations in $R^N$ and in $R_+^N$,, Proc. R. Soc. Edinb. Sect. A, 138 (2008), 339. doi: 10.1017/S0308210506000394. Google Scholar

[22]

B. Gidas, W. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n$,, in, 7a (1981). Google Scholar

[23]

B. Gidas and J. Spruck, A priori bounds for positive solutiions of nonlinear elliptic equations,, Comm. PDEs, 6 (1981), 883. doi: 10.1080/03605308108820196. Google Scholar

[24]

F. Hang, On the integral systems related to Hardy-Littlewood-Sobolev inequality,, Math. Res. Lett., 14 (2007), 373. Google Scholar

[25]

C. Jin and C. Li, Symmetry of solutions to some systems of integral equations,, Proc. AMS, 134 (2006), 1661. doi: 10.1090/S0002-9939-05-08411-X. Google Scholar

[26]

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

[27]

D. Li, G. Ströhmer and L. Wang, Symmetry of integral equations on bounded domains,, Proc. AMS, 137 (2009), 3695. doi: 10.1090/S0002-9939-09-09987-0. Google Scholar

[28]

Y. Y. Li, Remarks on some conformally invariant integral equations: The method of moving spheres,, J. Euro. Math. Soc., 6 (2004), 153. Google Scholar

[29]

C. Li and L. Ma, Uniqueness of positive bound states to Shrodinger systems with critical exponents,, SIAM J. of Appl. Anal., 40 (2008), 1049. doi: 10.1137/080712301. Google Scholar

[30]

C. Liu and S. Qiao, Symmetry and monotonicity for a system of integral equations,, Comm. Pure Appl. Anal., 8 (2009), 1925. doi: 10.3934/cpaa.2009.8.1925. Google Scholar

[31]

C.-S. Lin, A classification of solutions of a conformally invariant fourth order equation in $R^n$,, Comment. Math. Helv., 73 (1998), 206. doi: 10.1007/s000140050052. Google Scholar

[32]

S. Liu, Regularity, symmetry, and uniqueness of some integral type quasilinear equations,, Nonl. Anal., 71 (2009), 1796. doi: 10.1016/j.na.2009.01.014. Google Scholar

[33]

Y. Li and M. Zhu, Uniqueness theorems through the method of moving spheres,, Duke math. J., 80 (1995), 383. doi: 10.1215/S0012-7094-95-08016-8. Google Scholar

[34]

D. Li and R. Zhuo, An integral equation on half space,, Proc. AMS, 138 (2010), 2779. doi: 10.1090/S0002-9939-10-10368-2. Google Scholar

[35]

G. Lu and J. Zhu, Axial symmetry and regularity of solutions to an integral equation in a half space,, Pacific J. Math., 253 (2011), 455. doi: 10.2140/pjm.2011.253.455. Google Scholar

[36]

C. Ma, W. Chen and C. Li, Regularity of solutions for an integral system of Wolff type,, Adv. Math., 226 (2011), 2676. doi: 10.1016/j.aim.2010.07.020. Google Scholar

[37]

L. Ma and D. Chen, A Liouville type theorem for an integral system,, Commun. Pure Appl. Anal., 5 (2006), 855. doi: 10.3934/cpaa.2006.5.855. Google Scholar

[38]

L. Ma and B. Liu, Symmetry results for decay solutions of elliptic systems in the whole space,, Adv. Math., 225 (2010), 3052. doi: 10.1016/j.aim.2010.05.022. Google Scholar

[39]

L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation,, Arch. Rat. Mech. Anal., 195 (2010), 455. doi: 10.1007/s00205-008-0208-3. Google Scholar

[40]

W. Reichel and T. Weth, A priori bounds and a Liouville theorem on a half-space for higher-order elliptic Dirichlet problems,, Math. Z., 261 (2009), 805. doi: 10.1007/s00209-008-0352-3. Google Scholar

[41]

J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations,, Math. Ann., 313 (1999), 207. doi: 10.1007/s002080050258. Google Scholar

[1]

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

[2]

Weiwei Zhao, Jinge Yang, Sining Zheng. Liouville type theorem to an integral system in the half-space. Communications on Pure & Applied Analysis, 2014, 13 (2) : 511-525. doi: 10.3934/cpaa.2014.13.511

[3]

Dezhong Chen, Li Ma. A Liouville type Theorem for an integral system. Communications on Pure & Applied Analysis, 2006, 5 (4) : 855-859. doi: 10.3934/cpaa.2006.5.855

[4]

Jingbo Dou, Ye Li. Liouville theorem for an integral system on the upper half space. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 155-171. doi: 10.3934/dcds.2015.35.155

[5]

Xiaohui Yu. Liouville type theorems for singular integral equations and integral systems. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1825-1840. doi: 10.3934/cpaa.2016017

[6]

Anh Tuan Duong, Quoc Hung Phan. A Liouville-type theorem for cooperative parabolic systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 823-833. doi: 10.3934/dcds.2018035

[7]

Zhenjie Li, Ze Cheng, Dongsheng Li. The Liouville type theorem and local regularity results for nonlinear differential and integral systems. Communications on Pure & Applied Analysis, 2015, 14 (2) : 565-576. doi: 10.3934/cpaa.2015.14.565

[8]

Wei Dai, Zhao Liu, Guozhen Lu. Hardy-Sobolev type integral systems with Dirichlet boundary conditions in a half space. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1253-1264. doi: 10.3934/cpaa.2017061

[9]

Frank Arthur, Xiaodong Yan. A Liouville-type theorem for higher order elliptic systems of Hé non-Lane-Emden type. Communications on Pure & Applied Analysis, 2016, 15 (3) : 807-830. doi: 10.3934/cpaa.2016.15.807

[10]

Frank Arthur, Xiaodong Yan, Mingfeng Zhao. A Liouville-type theorem for higher order elliptic systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3317-3339. doi: 10.3934/dcds.2014.34.3317

[11]

Dorina Mitrea and Marius Mitrea. Boundary integral methods for harmonic differential forms in Lipschitz domains. Electronic Research Announcements, 1996, 2: 92-97.

[12]

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

[13]

Shigeru Sakaguchi. A Liouville-type theorem for some Weingarten hypersurfaces. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 887-895. doi: 10.3934/dcdss.2011.4.887

[14]

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

[15]

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

[16]

António J.G. Bento, Nicolae Lupa, Mihail Megan, César M. Silva. Integral conditions for nonuniform $μ$-dichotomy on the half-line. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3063-3077. doi: 10.3934/dcdsb.2017163

[17]

Keisuke Matsuya, Tetsuji Tokihiro. Existence and non-existence of global solutions for a discrete semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 209-220. doi: 10.3934/dcds.2011.31.209

[18]

Kaouther Ammar, Philippe Souplet. Liouville-type theorems and universal bounds for nonnegative solutions of the porous medium equation with source. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 665-689. doi: 10.3934/dcds.2010.26.665

[19]

Fuqin Sun, Mingxin Wang. Non-existence of global solutions for nonlinear strongly damped hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2005, 12 (5) : 949-958. doi: 10.3934/dcds.2005.12.949

[20]

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

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (19)
  • HTML views (0)
  • Cited by (16)

Other articles
by authors

[Back to Top]