American Institute of Mathematical Sciences

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

Liouville type theorems for poly-harmonic Navier problems

Linfen Cao 1, and  Wenxiong Chen 2,

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.
Keywords: ploy-harmonic equation, half spaces, Liouville-type theorem, systems of integral equations, non-existence of solutions., Kelvin transforms, Navier boundary conditions, method of moving planes in integral forms, equivalence.
Mathematics Subject Classification: Primary: 35J40, 35J61, 35B53; Secondary: 35J91, 35B0.
Citation: Linfen Cao, Wenxiong Chen. Liouville type theorems for poly-harmonic Navier problems. Discrete & Continuous Dynamical Systems, 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), 1671-1690. 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), 1-37. 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), 1365-1373. doi: 10.1016/j.jmaa.2012.01.015.  Google Scholar

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W. Chen and C. Li, "Methods on Nonlinear Elliptic Equations," AIMS Book Series on Diff. Equa. & Dyn. Sys., 4, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 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), 949-960. 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), 1167-1184. 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), 330-343. 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), 164-172.  Google Scholar

[9]

W. Chen, C. Li and B. Ou, Qualitative problems of solutions for an integral equation, Disc. Cont. Dyn. Sys., 12 (2005), 347-354.  Google Scholar

[10]

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.  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), 1-13. 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), 547-564. 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), 615-622. 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), 219-245. 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). 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), 91-102.  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), 744-753. 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), 2835-2867. 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), 663-678.  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), 339-359. 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 "Mathematical Analysis and Applications, Part A," Advances in Mathematics Suppl. Stud., 7a, Academic Press, New York-London, 1981.  Google Scholar

[23]

B. Gidas and J. Spruck, A priori bounds for positive solutiions of nonlinear elliptic equations, Comm. PDEs, 6 (1981), 883-901. doi: 10.1080/03605308108820196.  Google Scholar

[24]

F. Hang, On the integral systems related to Hardy-Littlewood-Sobolev inequality, Math. Res. Lett., 14 (2007), 373-383.  Google Scholar

[25]

C. Jin and C. Li, Symmetry of solutions to some systems of integral equations, Proc. AMS, 134 (2006), 1661-1670. 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), 221-231. 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), 3695-3702. 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), 153-180.  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), 1049-1057. 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), 1925-1932. 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), 206-231. doi: 10.1007/s000140050052.  Google Scholar

[32]

S. Liu, Regularity, symmetry, and uniqueness of some integral type quasilinear equations, Nonl. Anal., 71 (2009), 1796-1806. 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), 383-417. 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), 2779-2791. 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), 455-473. 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), 2676-2699. 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), 855-859. 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), 3052-3063. 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), 455-467. 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), 805-827. 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), 207-228. 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), 1671-1690. 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), 1-37. 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), 1365-1373. 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, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 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), 949-960. 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), 1167-1184. 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), 330-343. 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), 164-172.  Google Scholar

[9]

W. Chen, C. Li and B. Ou, Qualitative problems of solutions for an integral equation, Disc. Cont. Dyn. Sys., 12 (2005), 347-354.  Google Scholar

[10]

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.  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), 1-13. 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), 547-564. 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), 615-622. 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), 219-245. 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). 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), 91-102.  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), 744-753. 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), 2835-2867. 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), 663-678.  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), 339-359. 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 "Mathematical Analysis and Applications, Part A," Advances in Mathematics Suppl. Stud., 7a, Academic Press, New York-London, 1981.  Google Scholar

[23]

B. Gidas and J. Spruck, A priori bounds for positive solutiions of nonlinear elliptic equations, Comm. PDEs, 6 (1981), 883-901. doi: 10.1080/03605308108820196.  Google Scholar

[24]

F. Hang, On the integral systems related to Hardy-Littlewood-Sobolev inequality, Math. Res. Lett., 14 (2007), 373-383.  Google Scholar

[25]

C. Jin and C. Li, Symmetry of solutions to some systems of integral equations, Proc. AMS, 134 (2006), 1661-1670. 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), 221-231. 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), 3695-3702. 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), 153-180.  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), 1049-1057. 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), 1925-1932. 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), 206-231. doi: 10.1007/s000140050052.  Google Scholar

[32]

S. Liu, Regularity, symmetry, and uniqueness of some integral type quasilinear equations, Nonl. Anal., 71 (2009), 1796-1806. 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), 383-417. 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), 2779-2791. 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), 455-473. 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), 2676-2699. 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), 855-859. 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), 3052-3063. 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), 455-467. 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), 805-827. 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), 207-228. doi: 10.1007/s002080050258.  Google Scholar

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