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 and Applied Analysis, 2016, 15 (3) : 701-713. doi: 10.3934/cpaa.2016.15.701
References:
 [1] H. Berestycki, L. A. Caffarelli and L. Nireberg, Further qualitative properties for elliptic equations in unbounded domains, Ann. Norm. Sup. Pisa. C1, 25 (1997), 69-94. [2] G. Bianchi, Nonexistence of positive solutions to semilinear elliptic equations on $\mathbb{R^N}$ or $\mathbbR^N_+$ through the method of moving planes, Comm. in P.D.E., 22 (1997), 1671-1690. doi: 10.1080/03605309708821315. [3] T. Branson, S. Y. A. Chang and P. C. Yang, Estimates and extremal problems for the log-determinant on 4-manifolds, Commun. Math. Phys., 149 (1992), 241-262. [4] L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behaviour of semilinear elliptic equations with critical Sobolev growth, Comm. Pure App. Math., XLII (1989), 271-297. doi: 10.1002/cpa.3160420304. [5] W. Chen and C. Li, Calssification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622. doi: 10.1215/S0012-7094-91-06325-8. [6] E. Colorado Heras and I. Peral Alonso, Semilinear elliptic problems with mixed boundary conditions, J. Funct. Anal., 199 (2003), 468-507. doi: 10.1016/S0022-1236(02)00101-5. [7] L. Damascelli and F. Gladiali, Some nonexistence results for positive solutions of elliptic equations in unbounded domains, Rev. Mat. Iberoamericana, 20 (2004), 67-86. [8] E. N. Dancer, Some notes on the method of moving planes, Bull. Austral. Math. Soc., 46 (1992), 40-64. doi: 10.1017/S0004972700012089. [9] E. N. Dancer, Moving plane methods for system on half spaces, Math. Ann., 342 (2008), 245-254. doi: 10.1007/s00208-008-0226-3. [10] D. G. de Figueiredo, Semilimear Elliptic Systems, Research Surve, Universidade Estadual de Campinas, 1998. [11] D. G. de Figueiredo and B. Sirakov, Liouville type thoerems, monotonicity resluts and a prior bounds for positive solutions of elliptic system, Math. Ann., 333 (2005), 231-260. doi: 10.1007/s00208-005-0639-1. [12] B. Gidas, W. Ni and L. Nirenberg, Symmetry and related properties via maximum principle, Commun. Math. Phys., 68 (1979), 525-598. [13] B. Gidas and J. Spruk, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 24 (1981), 525-598. doi: 10.1002/cpa.3160340406. [14] B. Gidas and J. Spruk, A Priori bounds for positive solutions of nonlinear elliptic equations, Comm. in P.D.E., 6 (1981), 883-901. doi: 10.1080/03605308108820196. [15] Y. Guo and B. Li, Infinitely many solutions for the prescribed curvature problem of polyharmonic operator, Calc. Var. P.D.E., 46 (2013), 819-836. doi: 10.1007/s00526-012-0504-5. [16] Y. Guo, B. Li and J. Wei, Large energy entire solutions for the Yamabe type problem of polyharmonic operator, J. Diff. Equa., 254 (2013), 199-228. doi: 10.1016/j.jde.2012.08.038. [17] Y. Li and M. Zhu, Uniqueness theorem through the method of moving spheres, Duke Math. J., 80 (1995), 383-417. doi: 10.1215/S0012-7094-95-08016-8. [18] C. S. Lin, A classification of solutions of a conformally invariant fourth order equation in $\mathbb{R}^N2$, Comment. Math. Helv, 73 (1998), 206-231. doi: 10.1007/s000140050052. [19] P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, Parts I and II, Ann Inst. H. Poincare Analyse Nonlinear, 1 (1984), 109-2145 and 223-283. [20] J. Q. Liu, Y. X. Guo and Y. J. Zhang, Liouville type theorems for polyharmonic systems in $R^N$, Journal of Diff. Equ., 225 (2006), 685-709. doi: 10.1016/j.jde.2005.10.016. [21] W. Reichel and T. Weth, A prior 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. [22] S. Terracini, Symmetry properties of positives solutions to some elliptic equations with nonlinear boundary conditions, Diff. Int. Eq., 8 (1995), 1911-1922. [23] S. Terracini, On positive entire solutions to a class of equations with a singular coefficient and critical exponent, Adv. Diff. Eq., 1 (1996), 241-246. [24] J. Wei and X. Xu, Classification of solutions of higher oeder conformally invariant equations, Math. Ann., 313 (1999), 207-228. doi: 10.1007/s002080050258. [25] C. Wolf, A mathematical model for the propagation of a hantavirus in structured populations, Discrete Continuous Dynam. Systems-B, 4 (2004), 1065-1089. doi: 10.3934/dcdsb.2004.4.1065. [26] X. Xu, Uniqueness theorem for the entire positive solutions of biharmonic equations in $\mathbb{R}^N2$, Proc. Royal Soc. Edinburgh, 130A (2000), 651-670. doi: 10.1017/S0308210500000354.

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References:
 [1] H. Berestycki, L. A. Caffarelli and L. Nireberg, Further qualitative properties for elliptic equations in unbounded domains, Ann. Norm. Sup. Pisa. C1, 25 (1997), 69-94. [2] G. Bianchi, Nonexistence of positive solutions to semilinear elliptic equations on $\mathbb{R^N}$ or $\mathbbR^N_+$ through the method of moving planes, Comm. in P.D.E., 22 (1997), 1671-1690. doi: 10.1080/03605309708821315. [3] T. Branson, S. Y. A. Chang and P. C. Yang, Estimates and extremal problems for the log-determinant on 4-manifolds, Commun. Math. Phys., 149 (1992), 241-262. [4] L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behaviour of semilinear elliptic equations with critical Sobolev growth, Comm. Pure App. Math., XLII (1989), 271-297. doi: 10.1002/cpa.3160420304. [5] W. Chen and C. Li, Calssification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622. doi: 10.1215/S0012-7094-91-06325-8. [6] E. Colorado Heras and I. Peral Alonso, Semilinear elliptic problems with mixed boundary conditions, J. Funct. Anal., 199 (2003), 468-507. doi: 10.1016/S0022-1236(02)00101-5. [7] L. Damascelli and F. Gladiali, Some nonexistence results for positive solutions of elliptic equations in unbounded domains, Rev. Mat. Iberoamericana, 20 (2004), 67-86. [8] E. N. Dancer, Some notes on the method of moving planes, Bull. Austral. Math. Soc., 46 (1992), 40-64. doi: 10.1017/S0004972700012089. [9] E. N. Dancer, Moving plane methods for system on half spaces, Math. Ann., 342 (2008), 245-254. doi: 10.1007/s00208-008-0226-3. [10] D. G. de Figueiredo, Semilimear Elliptic Systems, Research Surve, Universidade Estadual de Campinas, 1998. [11] D. G. de Figueiredo and B. Sirakov, Liouville type thoerems, monotonicity resluts and a prior bounds for positive solutions of elliptic system, Math. Ann., 333 (2005), 231-260. doi: 10.1007/s00208-005-0639-1. [12] B. Gidas, W. Ni and L. Nirenberg, Symmetry and related properties via maximum principle, Commun. Math. Phys., 68 (1979), 525-598. [13] B. Gidas and J. Spruk, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 24 (1981), 525-598. doi: 10.1002/cpa.3160340406. [14] B. Gidas and J. Spruk, A Priori bounds for positive solutions of nonlinear elliptic equations, Comm. in P.D.E., 6 (1981), 883-901. doi: 10.1080/03605308108820196. [15] Y. Guo and B. Li, Infinitely many solutions for the prescribed curvature problem of polyharmonic operator, Calc. Var. P.D.E., 46 (2013), 819-836. doi: 10.1007/s00526-012-0504-5. [16] Y. Guo, B. Li and J. Wei, Large energy entire solutions for the Yamabe type problem of polyharmonic operator, J. Diff. Equa., 254 (2013), 199-228. doi: 10.1016/j.jde.2012.08.038. [17] Y. Li and M. Zhu, Uniqueness theorem through the method of moving spheres, Duke Math. J., 80 (1995), 383-417. doi: 10.1215/S0012-7094-95-08016-8. [18] C. S. Lin, A classification of solutions of a conformally invariant fourth order equation in $\mathbb{R}^N2$, Comment. Math. Helv, 73 (1998), 206-231. doi: 10.1007/s000140050052. [19] P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, Parts I and II, Ann Inst. H. Poincare Analyse Nonlinear, 1 (1984), 109-2145 and 223-283. [20] J. Q. Liu, Y. X. Guo and Y. J. Zhang, Liouville type theorems for polyharmonic systems in $R^N$, Journal of Diff. Equ., 225 (2006), 685-709. doi: 10.1016/j.jde.2005.10.016. [21] W. Reichel and T. Weth, A prior 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. [22] S. Terracini, Symmetry properties of positives solutions to some elliptic equations with nonlinear boundary conditions, Diff. Int. Eq., 8 (1995), 1911-1922. [23] S. Terracini, On positive entire solutions to a class of equations with a singular coefficient and critical exponent, Adv. Diff. Eq., 1 (1996), 241-246. [24] J. Wei and X. Xu, Classification of solutions of higher oeder conformally invariant equations, Math. Ann., 313 (1999), 207-228. doi: 10.1007/s002080050258. [25] C. Wolf, A mathematical model for the propagation of a hantavirus in structured populations, Discrete Continuous Dynam. Systems-B, 4 (2004), 1065-1089. doi: 10.3934/dcdsb.2004.4.1065. [26] X. Xu, Uniqueness theorem for the entire positive solutions of biharmonic equations in $\mathbb{R}^N2$, Proc. Royal Soc. Edinburgh, 130A (2000), 651-670. doi: 10.1017/S0308210500000354.
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