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Schrödinger-like operators and the eikonal equation
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 |
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. |
show all references
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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