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Regularity of the global attractor for a nonlinear Schrödinger equation with a point defect
Hardy-Sobolev type integral systems with Dirichlet boundary conditions in a half space
1. | chool of Mathematics and Systems Science, Beihang University (BUAA), Beijing, 100191, China |
2. | School of Mathematics and Computer Science, Jiangxi Science and Technology Normal University, Nanchang, 330038, China |
3. | Department of Mathematics, University of Connecticut, Storrs, CT 06269, USA |
In this paper, we investigate the Hardy-Sobolev type integral systems (1) with Dirichlet boundary conditions in a half space $\mathbb{R}_+^n$. We use the method of moving planes in integral forms introduced by Chen, Li and Ou [
References:
[1] |
G. Bianchi,
Non-existence of positive solutions to semilinear elliptic equations in ${\mathbb{R}^N}$ and $\mathbb{R}_N^ + $ through the method of moving plane, Comm. Partial Differential Equations, 22 (1997), 1671-1690.
doi: 10.1080/03605309708821315. |
[2] |
L. Cao and W. Chen,
Liouville type theorems for poly-harmonic Navier problems, Disc. Cont. Dyn. Sys., 33 (2013), 3937-3955.
doi: 10.3934/dcds.2013.33.3937. |
[3] |
L. Cao and Z. Dai,
A Liouville-type theorem for an integral equations system on a half-space $\mathbb{R}_n^ + $, J. Math. Anal. Appl., 389 (2012), 1365-1373.
doi: 10.1016/j.jmaa.2012.01.015. |
[4] |
G. Caristi, L. D'Ambrosio and E. Mitidieri,
Representation formulae for solutions to some classes of higher order systems and related Liouville theorems, Milan J. Math., 76 (2008), 27-67.
doi: 10.1007/s00032-008-0090-3. |
[5] |
W. Chen and Y. Fang,
Higher order or fractional order Hardy-Sobolev type equations, Bulletin of the Institute of Mathematics Academia Sinica (New Series), 9 (2014), 317-349.
|
[6] |
L. Caffarelli, B. Gidas and J. Spruck,
Asymptotic symmetry and local behavior of semilinear elliptic equation with critical Sobolev growth, Comm. Pure Appl. Math., 42 (1989), 271-297.
doi: 10.1002/cpa.3160420304. |
[7] |
D. Cao and Y. Li,
Results on positive solutions of elliptic equations with a critical HardySobolev operator, Methods Appl. Anal., 15 (2008), 81-96.
doi: 10.4310/MAA.2008.v15.n1.a8. |
[8] |
L. Chen, Z. Liu and G. Lu, Symmetry and regularity of solutions to the weighted HardySobolev type system, Adv. Nonlinear Stud. , 16 (2016), no. 1, 1-13.
doi: 10.1515/ans-2015-5005. |
[9] |
W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, AIMS Book Series on Diff. Equa. and Dyn. Sys. , Vol. 4,2010. |
[10] |
W. Chen and C. Li,
Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622.
doi: 10.1215/S0012-7094-91-06325-8. |
[11] |
W. Chen and C. Li,
An integral system and the Lane-Emden conjecture, Discrete Contin. Dyn. Syst., 4 (2009), 1167-1184.
doi: 10.3934/dcds.2009.24.1167. |
[12] |
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. |
[13] |
W. Chen, C. Li and B. Ou,
Classification of solutions for a system of integral equations, Comm. Patial Differential Equations, 30 (2005), 59-65.
doi: 10.1081/PDE-200044445. |
[14] |
W. Dai, Z. Liu and G. Lu, Liouville type theorems for PDE and IE systems involving fractional Laplacian on a half space, Potential Analysis, (2016).
doi: 10.1007/s11118-016-9594-6. |
[15] |
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. |
[16] |
Y. Fang and J. Zhang,
Nonexistece of positive solution for an integral equation on a half-space $\mathbb{R}_N^ + $, Comm. Pure Appl. Anal., 2 (2013), 663-678.
doi: 10.3934/cpaa.2013.12.663. |
[17] |
Y. Guo and J. Liu,
Liouville-type theorems for polyharmonic equations in ${\mathbb{R}^N}$ and in $\mathbb{R}_N^ + $, Proc. Roy. Edinburgh Sect. A., 138 (2008), 339-359.
doi: 10.1017/S0308210506000394. |
[18] |
B. Gidas, W. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in ${\mathbb{R}^N}$, Mathematical Analysis and Applications, vol. 7a of the book series Advances in Mathematics, Academic Press, New York, 1981. |
[19] |
B. Gidas and J. Spruck,
A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Patial Differential Equations, 6 (1981), 883-901.
doi: 10.1080/03605308108820196. |
[20] |
C. Jin and C. Li,
Symmetry of solution to some systems of integral equations, Proc. Amer. Math. Soc., 134 (2006), 1661-1670.
|
[21] |
E. Lieb,
Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. Math., 118 (1983), 349-374.
doi: 10.2307/2007032. |
[22] |
Z. Liu and W. Dai,
A Liouville type theorem for poly-harmonic system with Dirichlet boundary conditions in a half space, Advanced Nonlinear Studies, 15 (2015), 117-134.
doi: 10.1515/ans-2015-0106. |
[23] |
C. Li, A degree theory approach for the shooting method, preprint, arXiv: 1301.6232. |
[24] |
C. Li,
Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Invent. Math., 123 (1996), 221-231.
doi: 10.1007/s002220050023. |
[25] |
Y. Lei and C. Li, Decay properties of the Hardy-Littlewood-Sobolev systems of the LaneEmden type, preprint, arXiv: 1302.5567. |
[26] |
Y. Lei and C. Li,
Sharp criteria of Liouville type for some nonlinear systems, Discrete Contin. Dyn. Syst., 36 (2016), 3277-3315.
doi: 10.3934/dcds.2016.36.3277. |
[27] |
Y. Lei, C. Li and C. Ma,
Asymptotic radial symmetry and growth estimates of positive solutions to weighted Hardy-Littlewood-Sobolev system of integral equations, Calc. Var. and PDEs, 45 (2012), 43-61.
doi: 10.1007/s00526-011-0450-7. |
[28] |
D. Li, P. Niu and R. Zhuo,
Symmetry and non-existence of positive solutions for PDE system with Navier boundary conditions on a half space, Complex Variables and Elliptic Equations, 59 (2014), 1436-1450.
doi: 10.1080/17476933.2013.854346. |
[29] |
G. Lu, P. Wang and J. Zhu,
Liouville-type theorems and decay estimates for solutions to higher order elliptic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 653-665.
doi: 10.1016/j.anihpc.2012.02.004. |
[30] |
G. Lu and J. Zhu,
Symmetry and regularity of extremals of an integral equation related to the Hardy-Sobolev inequality, Calc. Var. and PDEs, 42 (2011), 563-577.
doi: 10.1007/s00526-011-0398-7. |
[31] |
G. Lu and J. Zhu,
Axial symmetry and regularity of solutions to an integral equations in a half-space, Pacific J. Math., 253 (2011), 455-473.
doi: 10.2140/pjm.2011.253.455. |
[32] |
G. Lu and J. Zhu,
Liouville-type theorems for fully nonlinear elliptic equations and systems in half spaces, Advanced Nonlinear Studies, 13 (2013), 979-1001.
doi: 10.1515/ans-2013-0413. |
[33] |
G. Lu and J. Zhu,
The maximum principles and symmetry results for viscosity solutions of fully nonlinear equations, J. Differ. Eqs, 258 (2015), 2054-2079.
doi: 10.1016/j.jde.2014.11.022. |
[34] |
L. Ma and D. Chen,
A Liouville type theorem for an intergral system, Comm. Pure Appl. Anal., 5 (2006), 855-859.
doi: 10.3934/cpaa.2006.5.855. |
[35] |
W. Reichel and T. Weth,
A prior bounds and a liouville theorem on a half-space for higherorder elliptic Dirichlet problems, Math. Z., 261 (2009), 805-827.
doi: 10.1007/s00209-008-0352-3. |
[36] |
J. Xu, H. Wu and Z. Tan,
Radial symmetry and asymptotic behaviors of positive solutions for certain nonlinear integral equations, J. Math. Anal. Appl., 427 (2015), 307-319.
doi: 10.1016/j.jmaa.2015.02.043. |
[37] |
J. Villavert,
Shooting with degree theory: Analysis of some weighted poly-harmonic systems, J. Differ. Eqs., 257 (2014), 1148-1167.
doi: 10.1016/j.jde.2014.05.003. |
[38] |
R. Zhuo, F. Li and B. Lv,
Liouville type theorems for Schrodinger system with Navier boundary conditions in a half space, Comm. Pure Appl. Anal., 3 (2014), 977-990.
doi: 10.3934/cpaa.2014.13.977. |
[39] |
Z. Zhang, Nonexistence of positive solutions of an integral system with weights, Adv. Difference Equ. , 61 (2011), 10 pp.
doi: 10.1186/1687-1847-2011-61. |
[40] |
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. |
show all references
References:
[1] |
G. Bianchi,
Non-existence of positive solutions to semilinear elliptic equations in ${\mathbb{R}^N}$ and $\mathbb{R}_N^ + $ through the method of moving plane, Comm. Partial Differential Equations, 22 (1997), 1671-1690.
doi: 10.1080/03605309708821315. |
[2] |
L. Cao and W. Chen,
Liouville type theorems for poly-harmonic Navier problems, Disc. Cont. Dyn. Sys., 33 (2013), 3937-3955.
doi: 10.3934/dcds.2013.33.3937. |
[3] |
L. Cao and Z. Dai,
A Liouville-type theorem for an integral equations system on a half-space $\mathbb{R}_n^ + $, J. Math. Anal. Appl., 389 (2012), 1365-1373.
doi: 10.1016/j.jmaa.2012.01.015. |
[4] |
G. Caristi, L. D'Ambrosio and E. Mitidieri,
Representation formulae for solutions to some classes of higher order systems and related Liouville theorems, Milan J. Math., 76 (2008), 27-67.
doi: 10.1007/s00032-008-0090-3. |
[5] |
W. Chen and Y. Fang,
Higher order or fractional order Hardy-Sobolev type equations, Bulletin of the Institute of Mathematics Academia Sinica (New Series), 9 (2014), 317-349.
|
[6] |
L. Caffarelli, B. Gidas and J. Spruck,
Asymptotic symmetry and local behavior of semilinear elliptic equation with critical Sobolev growth, Comm. Pure Appl. Math., 42 (1989), 271-297.
doi: 10.1002/cpa.3160420304. |
[7] |
D. Cao and Y. Li,
Results on positive solutions of elliptic equations with a critical HardySobolev operator, Methods Appl. Anal., 15 (2008), 81-96.
doi: 10.4310/MAA.2008.v15.n1.a8. |
[8] |
L. Chen, Z. Liu and G. Lu, Symmetry and regularity of solutions to the weighted HardySobolev type system, Adv. Nonlinear Stud. , 16 (2016), no. 1, 1-13.
doi: 10.1515/ans-2015-5005. |
[9] |
W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, AIMS Book Series on Diff. Equa. and Dyn. Sys. , Vol. 4,2010. |
[10] |
W. Chen and C. Li,
Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622.
doi: 10.1215/S0012-7094-91-06325-8. |
[11] |
W. Chen and C. Li,
An integral system and the Lane-Emden conjecture, Discrete Contin. Dyn. Syst., 4 (2009), 1167-1184.
doi: 10.3934/dcds.2009.24.1167. |
[12] |
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. |
[13] |
W. Chen, C. Li and B. Ou,
Classification of solutions for a system of integral equations, Comm. Patial Differential Equations, 30 (2005), 59-65.
doi: 10.1081/PDE-200044445. |
[14] |
W. Dai, Z. Liu and G. Lu, Liouville type theorems for PDE and IE systems involving fractional Laplacian on a half space, Potential Analysis, (2016).
doi: 10.1007/s11118-016-9594-6. |
[15] |
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. |
[16] |
Y. Fang and J. Zhang,
Nonexistece of positive solution for an integral equation on a half-space $\mathbb{R}_N^ + $, Comm. Pure Appl. Anal., 2 (2013), 663-678.
doi: 10.3934/cpaa.2013.12.663. |
[17] |
Y. Guo and J. Liu,
Liouville-type theorems for polyharmonic equations in ${\mathbb{R}^N}$ and in $\mathbb{R}_N^ + $, Proc. Roy. Edinburgh Sect. A., 138 (2008), 339-359.
doi: 10.1017/S0308210506000394. |
[18] |
B. Gidas, W. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in ${\mathbb{R}^N}$, Mathematical Analysis and Applications, vol. 7a of the book series Advances in Mathematics, Academic Press, New York, 1981. |
[19] |
B. Gidas and J. Spruck,
A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Patial Differential Equations, 6 (1981), 883-901.
doi: 10.1080/03605308108820196. |
[20] |
C. Jin and C. Li,
Symmetry of solution to some systems of integral equations, Proc. Amer. Math. Soc., 134 (2006), 1661-1670.
|
[21] |
E. Lieb,
Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. Math., 118 (1983), 349-374.
doi: 10.2307/2007032. |
[22] |
Z. Liu and W. Dai,
A Liouville type theorem for poly-harmonic system with Dirichlet boundary conditions in a half space, Advanced Nonlinear Studies, 15 (2015), 117-134.
doi: 10.1515/ans-2015-0106. |
[23] |
C. Li, A degree theory approach for the shooting method, preprint, arXiv: 1301.6232. |
[24] |
C. Li,
Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Invent. Math., 123 (1996), 221-231.
doi: 10.1007/s002220050023. |
[25] |
Y. Lei and C. Li, Decay properties of the Hardy-Littlewood-Sobolev systems of the LaneEmden type, preprint, arXiv: 1302.5567. |
[26] |
Y. Lei and C. Li,
Sharp criteria of Liouville type for some nonlinear systems, Discrete Contin. Dyn. Syst., 36 (2016), 3277-3315.
doi: 10.3934/dcds.2016.36.3277. |
[27] |
Y. Lei, C. Li and C. Ma,
Asymptotic radial symmetry and growth estimates of positive solutions to weighted Hardy-Littlewood-Sobolev system of integral equations, Calc. Var. and PDEs, 45 (2012), 43-61.
doi: 10.1007/s00526-011-0450-7. |
[28] |
D. Li, P. Niu and R. Zhuo,
Symmetry and non-existence of positive solutions for PDE system with Navier boundary conditions on a half space, Complex Variables and Elliptic Equations, 59 (2014), 1436-1450.
doi: 10.1080/17476933.2013.854346. |
[29] |
G. Lu, P. Wang and J. Zhu,
Liouville-type theorems and decay estimates for solutions to higher order elliptic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 653-665.
doi: 10.1016/j.anihpc.2012.02.004. |
[30] |
G. Lu and J. Zhu,
Symmetry and regularity of extremals of an integral equation related to the Hardy-Sobolev inequality, Calc. Var. and PDEs, 42 (2011), 563-577.
doi: 10.1007/s00526-011-0398-7. |
[31] |
G. Lu and J. Zhu,
Axial symmetry and regularity of solutions to an integral equations in a half-space, Pacific J. Math., 253 (2011), 455-473.
doi: 10.2140/pjm.2011.253.455. |
[32] |
G. Lu and J. Zhu,
Liouville-type theorems for fully nonlinear elliptic equations and systems in half spaces, Advanced Nonlinear Studies, 13 (2013), 979-1001.
doi: 10.1515/ans-2013-0413. |
[33] |
G. Lu and J. Zhu,
The maximum principles and symmetry results for viscosity solutions of fully nonlinear equations, J. Differ. Eqs, 258 (2015), 2054-2079.
doi: 10.1016/j.jde.2014.11.022. |
[34] |
L. Ma and D. Chen,
A Liouville type theorem for an intergral system, Comm. Pure Appl. Anal., 5 (2006), 855-859.
doi: 10.3934/cpaa.2006.5.855. |
[35] |
W. Reichel and T. Weth,
A prior bounds and a liouville theorem on a half-space for higherorder elliptic Dirichlet problems, Math. Z., 261 (2009), 805-827.
doi: 10.1007/s00209-008-0352-3. |
[36] |
J. Xu, H. Wu and Z. Tan,
Radial symmetry and asymptotic behaviors of positive solutions for certain nonlinear integral equations, J. Math. Anal. Appl., 427 (2015), 307-319.
doi: 10.1016/j.jmaa.2015.02.043. |
[37] |
J. Villavert,
Shooting with degree theory: Analysis of some weighted poly-harmonic systems, J. Differ. Eqs., 257 (2014), 1148-1167.
doi: 10.1016/j.jde.2014.05.003. |
[38] |
R. Zhuo, F. Li and B. Lv,
Liouville type theorems for Schrodinger system with Navier boundary conditions in a half space, Comm. Pure Appl. Anal., 3 (2014), 977-990.
doi: 10.3934/cpaa.2014.13.977. |
[39] |
Z. Zhang, Nonexistence of positive solutions of an integral system with weights, Adv. Difference Equ. , 61 (2011), 10 pp.
doi: 10.1186/1687-1847-2011-61. |
[40] |
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. |
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