# American Institute of Mathematical Sciences

May  2016, 15(3): 701-713. doi: 10.3934/cpaa.2016.15.701

## Nonexistence of positive solutions for polyharmonic systems in $\mathbb{R}^N_+$

 1 Department of Mathematics, Tsinghua University, Beijing, 100084 2 Department of Mathematics Science, Tsinghua University, Beijing 100084, China

Received  February 2014 Revised  October 2014 Published  February 2016

In this paper, we study the monotonicity and nonexistence of positive solutions for polyharmonic systems $\left\{\begin{array}{rlll} (-\Delta)^m u&=&f(u, v)\\ (-\Delta)^m v&=&g(u, v) \end{array}\right.\;\hbox{in}\;\mathbb{R}^N_+.$ By using the Alexandrov-Serrin method of moving plane combined with integral inequalities and Sobolev's inequality in a narrow domain, we prove the monotonicity of positive solutions for semilinear polyharmonic systems in $\mathbb{R_+^N}.$ As a result, the nonexistence for positive weak solutions to the system are obtained.
Citation: Yuxia Guo, Bo Li. Nonexistence of positive solutions for polyharmonic systems in $\mathbb{R}^N_+$. Communications on Pure & Applied Analysis, 2016, 15 (3) : 701-713. doi: 10.3934/cpaa.2016.15.701
##### References:
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##### References:
 [1] H. Berestycki, L. A. Caffarelli and L. Nireberg, Further qualitative properties for elliptic equations in unbounded domains,, \emph{Ann. Norm. Sup. Pisa. C1}, 25 (1997), 69. Google Scholar [2] G. Bianchi, Nonexistence of positive solutions to semilinear elliptic equations on $\mathbb{R^N}$ or $\mathbbR^N_+$ through the method of moving planes,, \emph{Comm. in P.D.E.}, 22 (1997), 1671. doi: 10.1080/03605309708821315. Google Scholar [3] T. Branson, S. Y. A. Chang and P. C. Yang, Estimates and extremal problems for the log-determinant on 4-manifolds,, \emph{Commun. Math. Phys.}, 149 (1992), 241. Google Scholar [4] L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behaviour of semilinear elliptic equations with critical Sobolev growth,, \emph{Comm. Pure App. Math.}, XLII (1989), 271. doi: 10.1002/cpa.3160420304. Google Scholar [5] W. Chen and C. Li, Calssification of solutions of some nonlinear elliptic equations,, \emph{Duke Math. J.}, 63 (1991), 615. doi: 10.1215/S0012-7094-91-06325-8. Google Scholar [6] E. Colorado Heras and I. Peral Alonso, Semilinear elliptic problems with mixed boundary conditions,, \emph{J. Funct. Anal.}, 199 (2003), 468. doi: 10.1016/S0022-1236(02)00101-5. Google Scholar [7] L. Damascelli and F. Gladiali, Some nonexistence results for positive solutions of elliptic equations in unbounded domains,, \emph{Rev. Mat. Iberoamericana}, 20 (2004), 67. Google Scholar [8] E. N. Dancer, Some notes on the method of moving planes,, \emph{Bull. Austral. Math. Soc.}, 46 (1992), 40. doi: 10.1017/S0004972700012089. Google Scholar [9] E. N. Dancer, Moving plane methods for system on half spaces,, \emph{Math. Ann.}, 342 (2008), 245. doi: 10.1007/s00208-008-0226-3. Google Scholar [10] D. G. de Figueiredo, Semilimear Elliptic Systems, Research Surve,, Universidade Estadual de Campinas, (1998). Google Scholar [11] D. G. de Figueiredo and B. Sirakov, Liouville type thoerems, monotonicity resluts and a prior bounds for positive solutions of elliptic system,, \emph{Math. Ann.}, 333 (2005), 231. doi: 10.1007/s00208-005-0639-1. Google Scholar [12] B. Gidas, W. Ni and L. Nirenberg, Symmetry and related properties via maximum principle,, \emph{Commun. Math. Phys.}, 68 (1979), 525. Google Scholar [13] B. Gidas and J. Spruk, Global and local behavior of positive solutions of nonlinear elliptic equations,, \emph{Comm. Pure Appl. Math.}, 24 (1981), 525. doi: 10.1002/cpa.3160340406. Google Scholar [14] B. Gidas and J. Spruk, A Priori bounds for positive solutions of nonlinear elliptic equations,, \emph{Comm. in P.D.E.}, 6 (1981), 883. doi: 10.1080/03605308108820196. Google Scholar [15] Y. Guo and B. Li, Infinitely many solutions for the prescribed curvature problem of polyharmonic operator,, \emph{Calc. Var. P.D.E.}, 46 (2013), 819. doi: 10.1007/s00526-012-0504-5. Google Scholar [16] Y. Guo, B. Li and J. Wei, Large energy entire solutions for the Yamabe type problem of polyharmonic operator,, \emph{J. Diff. Equa.}, 254 (2013), 199. doi: 10.1016/j.jde.2012.08.038. Google Scholar [17] Y. Li and M. Zhu, Uniqueness theorem through the method of moving spheres,, \emph{Duke Math. J.}, 80 (1995), 383. doi: 10.1215/S0012-7094-95-08016-8. Google Scholar [18] C. S. Lin, A classification of solutions of a conformally invariant fourth order equation in $\mathbbR^N$,, \emph{Comment. Math. Helv}, 73 (1998), 206. doi: 10.1007/s000140050052. Google Scholar [19] P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case,, \emph{Parts I and II, 1 (1984), 109. Google Scholar [20] J. Q. Liu, Y. X. Guo and Y. J. Zhang, Liouville type theorems for polyharmonic systems in $R^N$,, \emph{Journal of Diff. Equ.}, 225 (2006), 685. doi: 10.1016/j.jde.2005.10.016. Google Scholar [21] W. Reichel and T. Weth, A prior bounds and a Liouville theorem on a half space for higher-order elliptic Dirichlet problems,, \emph{Math. Z.}, 261 (2009), 805. doi: 10.1007/s00209-008-0352-3. Google Scholar [22] S. Terracini, Symmetry properties of positives solutions to some elliptic equations with nonlinear boundary conditions,, \emph{Diff. Int. Eq.}, 8 (1995), 1911. Google Scholar [23] S. Terracini, On positive entire solutions to a class of equations with a singular coefficient and critical exponent,, \emph{Adv. Diff. Eq.}, 1 (1996), 241. Google Scholar [24] J. Wei and X. Xu, Classification of solutions of higher oeder conformally invariant equations,, \emph{Math. Ann.}, 313 (1999), 207. doi: 10.1007/s002080050258. Google Scholar [25] C. Wolf, A mathematical model for the propagation of a hantavirus in structured populations,, \emph{Discrete Continuous Dynam. Systems-B}, 4 (2004), 1065. doi: 10.3934/dcdsb.2004.4.1065. Google Scholar [26] X. Xu, Uniqueness theorem for the entire positive solutions of biharmonic equations in $\mathbbR^N$,, \emph{Proc. Royal Soc. Edinburgh}, 130A (2000), 651. doi: 10.1017/S0308210500000354. Google Scholar
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