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A note on the Monge-Kantorovich problem in the plane
An integral equation involving Bessel potentials on half space
1. | Department of Mathematics, Harbin Institute of Technology, Harbin 150080, China |
2. | Science Research Center, Harbin Institute of Technology, Harbin, 150080 |
References:
[1] |
N. Aronszajn and K. T. Smith, Theory of Bessel potentials I, Ann. Inst. Fourier, 11 (1961), 385-475. |
[2] |
G. Bianchi, Non-existence of positive solutions to semilinear elliptic equations in $R^N$ or $\mathbbR^n_+$ through the method of moving planes, Comm. partial differential Equations, 22 (1997), 1671-1690.
doi: 10.1080/03605309708821315. |
[3] |
C. Coffman, Uniqueness of the ground state solution for $\Delta u-u+u^3=0$ and a variational characterization of other solutions, Arch. Rational Mech. Anal., 46 (1972), 81-95. |
[4] |
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. |
[5] |
L. Cao and Z, Dai, A Liouville-type theorem for an integral equation on a half-space $\mathbbR^n_+$, J. Math. Anal. Appl., 389 (2012), 1365-1373.
doi: 10.1016/j.jmaa.2012.01.015. |
[6] |
W. Chen, Y. Fang and C. Li, Super poly-harmonic property of solutions for Navier boundary problems in $\mathbbR^n_+$, J. Funct. Anal., 256 (2013), 1522-1555.
doi: 10.1016/j.jfa.2013.06.010. |
[7] |
W. Chen, Y. Fang and R. Yang, Semilinear equations involving the fractional Laplacian on domains,, arXiv:1309.7499v1 [math.AP]., ().
|
[8] |
W. Chen, C. Jin, C. Li and C. Lim, Weighted Hardy-Littlewood-Sobolev inequalities and system of integral equations, Disc. Cont. Dyn. Sys., S (2005), 164-172. |
[9] |
W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, AIMS Book Series on Diff. Equa. $&$ Dyn. Sys., 4 (2010). |
[10] |
W. Chen and C. Li, Regularity of solutions for a system of integral equations, Comm. Pure Appl. Anal., 4 (2005), 1-8. |
[11] |
W. Chen and C. Li, An integral system and the Lane-Emden conjecture, Disc. Cont. Dyn. Sys., 4 (2009), 1167-1184.
doi: 10.3934/dcds.2009.24.1167. |
[12] |
W. Chen and C. Li, Radial symmetry of solutions for some integral systems of Wolff type, Disc. Cont. Dyn. Sys., 30 (2011), 1083-1093.
doi: 10.3934/dcds.2011.30.1083. |
[13] |
W. Chen and C. Li, A sup + inf inequality near $R = 0$, Advances in Math, 220 (2009), 219-245.
doi: 10.1016/j.aim.2008.09.005. |
[14] |
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. |
[15] |
W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation, Disc. Cont. Dyn. Sys., 12 (2005), 347-354. |
[16] |
W. Chen and J. Zhu, Radial symmetry and regularity of solutions for poly-harmonic Dirichlet problems, J. Math. Anal. Appl., 377 (2011), 744-753.
doi: 10.1016/j.jmaa.2010.11.035. |
[17] |
L. Damascelli and F. Gladiali, Some nonexistence results for positive soluions of elliptic equantions in unbounded domain, Rev. Mat. Iberoam., 20 (2004), 67-86. |
[18] |
W. F. Donoghue, Distributions and Fourier Transforms, Vol. 32, Academic Press, Inc., New York, 1969. |
[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] |
B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations, 34 (1981), 525-598.
doi: 10.1080/03605308108820196. |
[21] |
L. Grafakos, Classical and Modern Fourier Analysis, Pearson Education, Inc., Upper Saddle River, NJ, 2004. |
[22] |
X. Han and G. Lu, Regularity of solutions to an integral equation associated with bessel potential, Comm. Pure Appl. Anal., 10 (2011), 1111-1119.
doi: 10.3934/cpaa.2011.10.1111. |
[23] |
M. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $R^n$, Arch. Rational Mech. Anal., 105 (1989), 243-266.
doi: 10.1007/BF00251502. |
[24] |
S. Liu, Regularity, symmetry, and uniqueness of some integral type quasilinear equations, Nonlinear Anal., 71 (2009), 4231-4240.
doi: 10.1016/j.na.2009.01.014. |
[25] |
C. Li and J. Lim, The singularity analysis of solutions to some integral equations, Comm. Pure Appl. Anal., 6 (2007), 453-464.
doi: 10.3934/cpaa.2007.6.453. |
[26] |
C. Liu and S. Qiao, Symmetry and monotonicity for a system of integral equations, Comm. Pure Appl. Anal., 6 (2009), 1925-1932.
doi: 10.3934/cpaa.2009.8.1925. |
[27] |
Y. Li and M. Zhu, Uniqueness theorems through the mothod of moving spheres, Duke Math. J., 80 (1995), 383-417.
doi: 10.1215/S0012-7094-95-08016-8. |
[28] |
L. Ma and D. Chen, Radial symmetry and monotonicity for an integral equation, J. Math. Anal. Appl., 342 (2008), 943-949.
doi: 10.1016/j.jmaa.2007.12.064. |
[29] |
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. |
[30] |
L. Ma and D. Chen, Radial symmetry and uniqueness for positive solutions of a Schrödinger type system, Math. Comput. Modelling, 49 (2009), 379-385.
doi: 10.1016/j.mcm.2008.06.010. |
[31] |
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. |
[32] |
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. |
[33] |
K. Mcleod and J. Serrin, Uniqueness of positive radial solutions of $\Delta u+f(u)=0$ in $R^n$, Arch. Rational Mech. Anal., 99 (1987), 115-145.
doi: 10.1007/BF00275874. |
[34] |
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. |
[35] |
E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, Vol. 30, Princeton University Press, Princeton, N.J., 1970. |
[36] |
Y. Zhao, Regularity and symmetry for solutions to a system of weighted integral equations, J. Math. Anal. Appl., 391 (2012), 209-222.
doi: 10.1016/j.jmaa.2012.02.016. |
[37] |
R. Zhuo, F. Li and B. Lv, Liouville type theorems for Schrödinger system with Navier boundary conditions in a half space, Comm. Pure Appl. Anal., 13 (2014), 977-990.
doi: 10.3934/cpaa.2014.13.977. |
[38] |
W. Ziemer, Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation, in: Graduate Texts in Mathematics, Vol. 120, Springer-Verlag, New York, 1989.
doi: 10.1007/978-1-4612-1015-3. |
show all references
References:
[1] |
N. Aronszajn and K. T. Smith, Theory of Bessel potentials I, Ann. Inst. Fourier, 11 (1961), 385-475. |
[2] |
G. Bianchi, Non-existence of positive solutions to semilinear elliptic equations in $R^N$ or $\mathbbR^n_+$ through the method of moving planes, Comm. partial differential Equations, 22 (1997), 1671-1690.
doi: 10.1080/03605309708821315. |
[3] |
C. Coffman, Uniqueness of the ground state solution for $\Delta u-u+u^3=0$ and a variational characterization of other solutions, Arch. Rational Mech. Anal., 46 (1972), 81-95. |
[4] |
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. |
[5] |
L. Cao and Z, Dai, A Liouville-type theorem for an integral equation on a half-space $\mathbbR^n_+$, J. Math. Anal. Appl., 389 (2012), 1365-1373.
doi: 10.1016/j.jmaa.2012.01.015. |
[6] |
W. Chen, Y. Fang and C. Li, Super poly-harmonic property of solutions for Navier boundary problems in $\mathbbR^n_+$, J. Funct. Anal., 256 (2013), 1522-1555.
doi: 10.1016/j.jfa.2013.06.010. |
[7] |
W. Chen, Y. Fang and R. Yang, Semilinear equations involving the fractional Laplacian on domains,, arXiv:1309.7499v1 [math.AP]., ().
|
[8] |
W. Chen, C. Jin, C. Li and C. Lim, Weighted Hardy-Littlewood-Sobolev inequalities and system of integral equations, Disc. Cont. Dyn. Sys., S (2005), 164-172. |
[9] |
W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, AIMS Book Series on Diff. Equa. $&$ Dyn. Sys., 4 (2010). |
[10] |
W. Chen and C. Li, Regularity of solutions for a system of integral equations, Comm. Pure Appl. Anal., 4 (2005), 1-8. |
[11] |
W. Chen and C. Li, An integral system and the Lane-Emden conjecture, Disc. Cont. Dyn. Sys., 4 (2009), 1167-1184.
doi: 10.3934/dcds.2009.24.1167. |
[12] |
W. Chen and C. Li, Radial symmetry of solutions for some integral systems of Wolff type, Disc. Cont. Dyn. Sys., 30 (2011), 1083-1093.
doi: 10.3934/dcds.2011.30.1083. |
[13] |
W. Chen and C. Li, A sup + inf inequality near $R = 0$, Advances in Math, 220 (2009), 219-245.
doi: 10.1016/j.aim.2008.09.005. |
[14] |
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. |
[15] |
W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation, Disc. Cont. Dyn. Sys., 12 (2005), 347-354. |
[16] |
W. Chen and J. Zhu, Radial symmetry and regularity of solutions for poly-harmonic Dirichlet problems, J. Math. Anal. Appl., 377 (2011), 744-753.
doi: 10.1016/j.jmaa.2010.11.035. |
[17] |
L. Damascelli and F. Gladiali, Some nonexistence results for positive soluions of elliptic equantions in unbounded domain, Rev. Mat. Iberoam., 20 (2004), 67-86. |
[18] |
W. F. Donoghue, Distributions and Fourier Transforms, Vol. 32, Academic Press, Inc., New York, 1969. |
[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] |
B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations, 34 (1981), 525-598.
doi: 10.1080/03605308108820196. |
[21] |
L. Grafakos, Classical and Modern Fourier Analysis, Pearson Education, Inc., Upper Saddle River, NJ, 2004. |
[22] |
X. Han and G. Lu, Regularity of solutions to an integral equation associated with bessel potential, Comm. Pure Appl. Anal., 10 (2011), 1111-1119.
doi: 10.3934/cpaa.2011.10.1111. |
[23] |
M. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $R^n$, Arch. Rational Mech. Anal., 105 (1989), 243-266.
doi: 10.1007/BF00251502. |
[24] |
S. Liu, Regularity, symmetry, and uniqueness of some integral type quasilinear equations, Nonlinear Anal., 71 (2009), 4231-4240.
doi: 10.1016/j.na.2009.01.014. |
[25] |
C. Li and J. Lim, The singularity analysis of solutions to some integral equations, Comm. Pure Appl. Anal., 6 (2007), 453-464.
doi: 10.3934/cpaa.2007.6.453. |
[26] |
C. Liu and S. Qiao, Symmetry and monotonicity for a system of integral equations, Comm. Pure Appl. Anal., 6 (2009), 1925-1932.
doi: 10.3934/cpaa.2009.8.1925. |
[27] |
Y. Li and M. Zhu, Uniqueness theorems through the mothod of moving spheres, Duke Math. J., 80 (1995), 383-417.
doi: 10.1215/S0012-7094-95-08016-8. |
[28] |
L. Ma and D. Chen, Radial symmetry and monotonicity for an integral equation, J. Math. Anal. Appl., 342 (2008), 943-949.
doi: 10.1016/j.jmaa.2007.12.064. |
[29] |
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. |
[30] |
L. Ma and D. Chen, Radial symmetry and uniqueness for positive solutions of a Schrödinger type system, Math. Comput. Modelling, 49 (2009), 379-385.
doi: 10.1016/j.mcm.2008.06.010. |
[31] |
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. |
[32] |
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. |
[33] |
K. Mcleod and J. Serrin, Uniqueness of positive radial solutions of $\Delta u+f(u)=0$ in $R^n$, Arch. Rational Mech. Anal., 99 (1987), 115-145.
doi: 10.1007/BF00275874. |
[34] |
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. |
[35] |
E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, Vol. 30, Princeton University Press, Princeton, N.J., 1970. |
[36] |
Y. Zhao, Regularity and symmetry for solutions to a system of weighted integral equations, J. Math. Anal. Appl., 391 (2012), 209-222.
doi: 10.1016/j.jmaa.2012.02.016. |
[37] |
R. Zhuo, F. Li and B. Lv, Liouville type theorems for Schrödinger system with Navier boundary conditions in a half space, Comm. Pure Appl. Anal., 13 (2014), 977-990.
doi: 10.3934/cpaa.2014.13.977. |
[38] |
W. Ziemer, Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation, in: Graduate Texts in Mathematics, Vol. 120, Springer-Verlag, New York, 1989.
doi: 10.1007/978-1-4612-1015-3. |
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