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Piecewise linear perturbations of a linear center
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 |
  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.
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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