Liouville type theorems for poly-harmonic Navier problems

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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

Abstract
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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 and 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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[9]	W. Chen, C. Li and B. Ou, Qualitative problems of solutions for an integral equation, Disc. Cont. Dyn. Sys., 12 (2005), 347-354.
[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.
[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).
[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.
[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.
[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.
[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.
[16]	Super polyharmonic property of solutions of Navier boundary problem in $R^n_+$, preprint, (2012).
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[24]	F. Hang, On the integral systems related to Hardy-Littlewood-Sobolev inequality, Math. Res. Lett., 14 (2007), 373-383.
[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.
[26]	C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Invent. Math., 123 (1996), 221-231. doi: 10.1007/s002220050023.
[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.
[28]	Y. Y. Li, Remarks on some conformally invariant integral equations: The method of moving spheres, J. Euro. Math. Soc., 6 (2004), 153-180.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[9]	W. Chen, C. Li and B. Ou, Qualitative problems of solutions for an integral equation, Disc. Cont. Dyn. Sys., 12 (2005), 347-354.
[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.
[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).
[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.
[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.
[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.
[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.
[16]	Super polyharmonic property of solutions of Navier boundary problem in $R^n_+$, preprint, (2012).
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[24]	F. Hang, On the integral systems related to Hardy-Littlewood-Sobolev inequality, Math. Res. Lett., 14 (2007), 373-383.
[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.
[26]	C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Invent. Math., 123 (1996), 221-231. doi: 10.1007/s002220050023.
[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.
[28]	Y. Y. Li, Remarks on some conformally invariant integral equations: The method of moving spheres, J. Euro. Math. Soc., 6 (2004), 153-180.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

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