# 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 & 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
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