# American Institute of Mathematical Sciences

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 $R^n_+$: $$\left\{\begin{array}{l} (-\triangle)^{\frac{\alpha}{2}} u=u^p,\ \ \ \ \ \:\:\: \:\:\:\:\:\ \:\:\ \ \ \ \ \ \ \ \ \ \ \ \:\:\:\:\ \mbox{in}\,\ R^n_+,\\ u=-\triangle u=\cdots=(-\triangle)^{\frac{\alpha}{2}-1}u=0, \ \ \ \mbox{on}\ \partial R^n_+, \end{array} \right. \label{phe1}$$ 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 $$u(x)=\int_{R^n_+}G(x,y,\alpha) u^p(y)dy,\,\,\,\,\, x\in\,R^n_+, \label{ie0}$$ 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 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.

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##### 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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