American Institute of Mathematical Sciences

March  2014, 13(2): 511-525. doi: 10.3934/cpaa.2014.13.511

Liouville type theorem to an integral system in the half-space

 1 School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China, China 2 Department of Applied Mathematics, Dalian University of Technology, Dalian 116024

Received  July 2012 Revised  June 2013 Published  October 2013

In this paper, by using the moving plane method in integral forms, we establish a Liouville type theorem for a coupled integral system with Navier boundary values in the half-space. Furthermore, we prove that the Liouville type theorem is valid for the related differential system as well under an additional assumption by showing the equivalence between the involved differential and integral systems.
Citation: Weiwei Zhao, Jinge Yang, Sining Zheng. Liouville type theorem to an integral system in the half-space. Communications on Pure and Applied Analysis, 2014, 13 (2) : 511-525. doi: 10.3934/cpaa.2014.13.511
References:
 [1] I. Birindelli and E. Mitidieri, Liouville theorems for elliptic inequalities and applications, Proc. Roy. Soc. Edinburgh Sect. A, 128 (1998), 1217-1247. doi: 10.1017/S0308210500027293. [2] T. Branson, Differential operators canonically associated to a conformal structure, Math. Scand., 2 (1985), 293-345. [3] L. Cao and W. Chen, Liouville type theorems for poly-harmonic Navier problems, Discrete Contin. Dyn. Syst., 33 (2013), 3937-3955. doi: 10.3934/dcds.2013.33.3937. [4] 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. [5] W. Chen, C. Jin and C. Li, Weighted Hardy-Littlewood-Sobolev inequalities and system of integral equations, Discrete Contin. Dyn. Syst., suppl. (2005), 164-172. [6] 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. [7] W. Chen and C. Li, Super polyharmonic property of solutions for PDE systems and its applications, Commun. Pure Appl. Anal., 12 (2013), 2497-2514. doi: 10.3934/cpaa.2013.12.2497. [8] W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343. doi: 10.1002/cpa.20116. [9] W. Chen, C. Li and G. Wang, On the stationary solutions of the 2D Doi-Onsager model, Nonlinear Anal., 73 (2010), 2410-2425. doi: 10.1016/j.na.2010.06.012. [10] Z. Chen and Z. Zhao, Potential theory for elliptic systems, Ann. Probab., 24 (1996), 293-319. doi: 10.1214/aop/1042644718. [11] Z. Djadli, A. Malchiodi and M. Almedou, Prescribing a fourth order conformal invariant on the standard sphere. II. Blow up analysis and applications, Ann. Sc. Norm. Super. Pisa Cl. Sci., 1 (2002), 387-434. [12] P. Esposito and F. Robert, Mountain pass critical points for Paneitz-Branson operators, Calc. Var. Partial Differential Equations, 15 (2002), 493-517. doi: 10.1007/s005260100134. [13] 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. [14] D. G. de Figueiredo and P. Felmer, A Liouville-type theorem for elliptic systems, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 21 (1994), 387-397. [15] X. Huang, D. Li and L. Wang, Symmetry and monotonicity for integral equation systems, Nonlinear Anal. Real World Appl., 12 (2011), 3515-3530. doi: 10.1016/j.nonrwa.2011.06.012. [16] C. Jin and C. Li, Symmetry of solutions to some systems of integral equations, Proc. Amer. Math. Soc., 134 (2006), 1661-1670 . doi: 10.1090/S0002-9939-05-08411-X. [17] Y. Lei and C. Ma, Radial symmetry and decay rates of positive solutions of a wolff type integral system, Proc. Amer. Math. Soc., 140 (2012), 541-551. doi: 10.1090/S0002-9939-2011-11401-1. [18] S. Lenhart and S. Belbas, A system of nonlinear partial differential equations arising in the optimal control of stochastic systems with switching costs, SIAM J. Appl. Math., 43 (1983), 465-475. doi: 10.1137/0143030. [19] C. Li and L. Ma, Uniqueness of positive bound states to shrödinger systems with critical exponents, SIAM J. Math. Anal., 40 (2008), 1049-1057. doi: 10.1137/080712301. [20] D. Li and R. Zhuo, An integral equation on half space, Proc. Amer. Math. Soc., 138 (2010), 2779-2791. doi: 10.1090/S0002-9939-10-10368-2. [21] J. Liu, Y. Guo and Y. Zhang, Liouville-type theorems for polyharmonic systems in $\mathbb{R}^{N}$, J. Differential Equations, 225 (2006), 685-709. doi: 10.1016/j.jde.2005.10.016. [22] 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. [23] E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $\mathbb{R}^{N}$, Differential Integral Equations, 9 (1996), 465-479. [24] S. Nazarov and G. Sweers, A hinged plate equation and iterated Dirichlet Laplace operator on domains with concave corners, J. Differential Equations, 233 (2007), 151-180. doi: 10.1016/j.jde.2006.09.018. [25] L. Peletier, Nonlinear eigenvalue problems for higher-order model equations, in "Handbook of Differential Equations, Stationary Partial Differential Equations," Volume 3, Chapter 7 (Eds. M. Chipot and P. Quittner), Elsevier, 2006. [26] L. Peletier and W. Troy, "Spatial Patterns. Higher Order Models in Physics and Mechanics, Progress in Nonlinear Differential Equations and Their Applications. 45," Birkhauser Boston, Inc., Boston, MA, 2001. [27] P. Peter, P. Quittner and Ph. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems. Part I: Elliptic equations and systems, Duke Math. J., 139 (2007), 555-579. doi: 10.1215/S0012-7094-07-13935-8. [28] 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. [29] W. Reichel and T. Weth, Existence of solutions to nonlinear, subcritical higher order elliptic Dirichlet problems, J. Differential Equations, 248 (2010), 1866-1878. doi: 10.1016/j.jde.2009.09.012. [30] J. Serrin and H. Zou, Nonexistence of positive solutions of Lane-Emden systems, Differential Integral Equations, 9 (1996), 635-653. [31] B. Sirakov, Existence results and a priori bounds for higher order elliptic equations and systems, J. Math. Pures Appl., 89 (2008), 114-133. doi: 10.1016/j.matpur.2007.10.003. [32] J.B. Van den Berg, The phase-plane picture for a class of fourth-order conservative differential equations, J. Differential Equations, 161 (2000), 110-153. doi: 10.1006/jdeq.1999.3698. [33] J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann., 313 (1999), 207-228. doi: 10.1017/S0308210500027293.

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References:
 [1] I. Birindelli and E. Mitidieri, Liouville theorems for elliptic inequalities and applications, Proc. Roy. Soc. Edinburgh Sect. A, 128 (1998), 1217-1247. doi: 10.1017/S0308210500027293. [2] T. Branson, Differential operators canonically associated to a conformal structure, Math. Scand., 2 (1985), 293-345. [3] L. Cao and W. Chen, Liouville type theorems for poly-harmonic Navier problems, Discrete Contin. Dyn. Syst., 33 (2013), 3937-3955. doi: 10.3934/dcds.2013.33.3937. [4] 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. [5] W. Chen, C. Jin and C. Li, Weighted Hardy-Littlewood-Sobolev inequalities and system of integral equations, Discrete Contin. Dyn. Syst., suppl. (2005), 164-172. [6] 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. [7] W. Chen and C. Li, Super polyharmonic property of solutions for PDE systems and its applications, Commun. Pure Appl. Anal., 12 (2013), 2497-2514. doi: 10.3934/cpaa.2013.12.2497. [8] W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343. doi: 10.1002/cpa.20116. [9] W. Chen, C. Li and G. Wang, On the stationary solutions of the 2D Doi-Onsager model, Nonlinear Anal., 73 (2010), 2410-2425. doi: 10.1016/j.na.2010.06.012. [10] Z. Chen and Z. Zhao, Potential theory for elliptic systems, Ann. Probab., 24 (1996), 293-319. doi: 10.1214/aop/1042644718. [11] Z. Djadli, A. Malchiodi and M. Almedou, Prescribing a fourth order conformal invariant on the standard sphere. II. Blow up analysis and applications, Ann. Sc. Norm. Super. Pisa Cl. Sci., 1 (2002), 387-434. [12] P. Esposito and F. Robert, Mountain pass critical points for Paneitz-Branson operators, Calc. Var. Partial Differential Equations, 15 (2002), 493-517. doi: 10.1007/s005260100134. [13] 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. [14] D. G. de Figueiredo and P. Felmer, A Liouville-type theorem for elliptic systems, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 21 (1994), 387-397. [15] X. Huang, D. Li and L. Wang, Symmetry and monotonicity for integral equation systems, Nonlinear Anal. Real World Appl., 12 (2011), 3515-3530. doi: 10.1016/j.nonrwa.2011.06.012. [16] C. Jin and C. Li, Symmetry of solutions to some systems of integral equations, Proc. Amer. Math. Soc., 134 (2006), 1661-1670 . doi: 10.1090/S0002-9939-05-08411-X. [17] Y. Lei and C. Ma, Radial symmetry and decay rates of positive solutions of a wolff type integral system, Proc. Amer. Math. Soc., 140 (2012), 541-551. doi: 10.1090/S0002-9939-2011-11401-1. [18] S. Lenhart and S. Belbas, A system of nonlinear partial differential equations arising in the optimal control of stochastic systems with switching costs, SIAM J. Appl. Math., 43 (1983), 465-475. doi: 10.1137/0143030. [19] C. Li and L. Ma, Uniqueness of positive bound states to shrödinger systems with critical exponents, SIAM J. Math. Anal., 40 (2008), 1049-1057. doi: 10.1137/080712301. [20] D. Li and R. Zhuo, An integral equation on half space, Proc. Amer. Math. Soc., 138 (2010), 2779-2791. doi: 10.1090/S0002-9939-10-10368-2. [21] J. Liu, Y. Guo and Y. Zhang, Liouville-type theorems for polyharmonic systems in $\mathbb{R}^{N}$, J. Differential Equations, 225 (2006), 685-709. doi: 10.1016/j.jde.2005.10.016. [22] 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. [23] E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $\mathbb{R}^{N}$, Differential Integral Equations, 9 (1996), 465-479. [24] S. Nazarov and G. Sweers, A hinged plate equation and iterated Dirichlet Laplace operator on domains with concave corners, J. Differential Equations, 233 (2007), 151-180. doi: 10.1016/j.jde.2006.09.018. [25] L. Peletier, Nonlinear eigenvalue problems for higher-order model equations, in "Handbook of Differential Equations, Stationary Partial Differential Equations," Volume 3, Chapter 7 (Eds. M. Chipot and P. Quittner), Elsevier, 2006. [26] L. Peletier and W. Troy, "Spatial Patterns. Higher Order Models in Physics and Mechanics, Progress in Nonlinear Differential Equations and Their Applications. 45," Birkhauser Boston, Inc., Boston, MA, 2001. [27] P. Peter, P. Quittner and Ph. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems. Part I: Elliptic equations and systems, Duke Math. J., 139 (2007), 555-579. doi: 10.1215/S0012-7094-07-13935-8. [28] 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. [29] W. Reichel and T. Weth, Existence of solutions to nonlinear, subcritical higher order elliptic Dirichlet problems, J. Differential Equations, 248 (2010), 1866-1878. doi: 10.1016/j.jde.2009.09.012. [30] J. Serrin and H. Zou, Nonexistence of positive solutions of Lane-Emden systems, Differential Integral Equations, 9 (1996), 635-653. [31] B. Sirakov, Existence results and a priori bounds for higher order elliptic equations and systems, J. Math. Pures Appl., 89 (2008), 114-133. doi: 10.1016/j.matpur.2007.10.003. [32] J.B. Van den Berg, The phase-plane picture for a class of fourth-order conservative differential equations, J. Differential Equations, 161 (2000), 110-153. doi: 10.1006/jdeq.1999.3698. [33] J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann., 313 (1999), 207-228. doi: 10.1017/S0308210500027293.
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