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Global bifurcations and a priori bounds of positive solutions for coupled nonlinear Schrödinger Systems
Least energy solutions for fractional Kirchhoff type equations involving critical growth
1. | School of Mathematics and information, Guangxi University, Nanning 530004, China |
2. | Department of Mathematics, Central China Normal University, Wuhan 430079, China |
3. | School of Science, East China JiaoTong University, Nanchang 330013, China |
$ \begin{equation*} \begin{array}{ll} \left \{ \begin{array}{ll} \Big(a+b\int_{ \mathbb{R} ^3}|(-\Delta)^\frac{s}{2}u|^2dx\Big)(-\Delta )^s u+V(x)u = f(u)+|u|^{2^*_s-2}u, \ x\in \mathbb{R} ^3, \\ u\in H^s( \mathbb{R} ^3), \end{array} \right . \end{array} \end{equation*} $ |
$ a, \ b>0 $ |
$ 2^*_s = \frac{6}{3-2s} $ |
$ s\in(0, 1) $ |
$ \mathbb{R} ^3 $ |
$ V $ |
$ \mathbb{R} ^3 $ |
$ f $ |
$ V $ |
References:
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C. O. Alves, F. J. S. A. Corrêa and T. F. Ma,
Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. Math. Appl., 49 (2005), 85-93.
doi: 10.1016/j.camwa.2005.01.008. |
[2] |
D. Applebaum,
Lévy processes-from probability to finance quantum groups, Notices Am. Math. Soc., 51 (2004), 1336-1347.
|
[3] |
A. Arosio and S. Panizzi,
On the well-posedness of the Kirchhoff string, Trans. Amer. Math. Soc., 348 (1996), 305-330.
doi: 10.1090/S0002-9947-96-01532-2. |
[4] |
P. Biler, G. Karch and W. A. Woyczyński,
Critical nonlinearity exponent and self-similar asymptotics for Lévy conservation laws, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18 (2001), 613-637.
doi: 10.1016/S0294-1449(01)00080-4. |
[5] |
C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, Lecture Notes of the Unione Matematica Italiana, 20. Springer, [Cham]; Unione Matematica Italiana, Bologna, 2016.
doi: 10.1007/978-3-319-28739-3. |
[6] |
L. Caffarelli, J. M. Roquejoffre and O. Savin,
Non-local minimal surfaces, Comm. Pure Appl. Math., 63 (2010), 1111-1144.
doi: 10.1002/cpa.20331. |
[7] |
L. Caffarelli and L. Silvestre,
An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[8] |
L. Caffarelli and E. Valdinoci,
Uniform estimates and limiting arguments for nonlocal minimal surfaces, Calc. Var. Partial Differential Equations, 41 (2011), 203-240.
doi: 10.1007/s00526-010-0359-6. |
[9] |
M. M. Cavalcanti, V. N. Domingos Cavalcanti and J. A. Soriano,
Global existence and uniform decay rates for the Kirchhoff-Carrier equation with nonlinear dissipation, Adv. Differential Equations, 6 (2001), 701-730.
|
[10] |
X. Chang and Z. Wang,
Ground state of scalar field equations involving a fractional Laplacian with general nonlinearity, Nonlinearity, 26 (2013), 479-494.
doi: 10.1088/0951-7715/26/2/479. |
[11] |
C. Chen, Y. Kuo and T. Wu,
The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions, J. Differential Equations, 250 (2011), 1876-1908.
doi: 10.1016/j.jde.2010.11.017. |
[12] |
R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman & Hall/CRC, Boca Raton, FL, 2004. |
[13] |
A. Cotsiolis and N. K. Tavoularis,
Best constants for Sobolev inequalities for higher order fractional derivatives, J. Math. Anal. Appl., 295 (2004), 225-236.
doi: 10.1016/j.jmaa.2004.03.034. |
[14] |
P. D'Ancona and S. Spagnolo,
Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. Math., 108 (1992), 247-262.
doi: 10.1007/BF02100605. |
[15] |
Y. Deng, S. Peng and W. Shuai,
Existence and asympototic behavior of nodal solutions for the Kirchhoff-type problems in $ \mathbb{R} ^3$, J. Funct. Anal., 269 (2015), 3500-3527.
doi: 10.1016/j.jfa.2015.09.012. |
[16] |
E. Di Nezza, G. Palatucci and E. Valdinoci,
Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.
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[17] |
S. Dipierro, M. Medina and E. Valdinoci, Fractional Elliptic Problems with Critical Growth in the Whole of $ \mathbb{R} ^n$, Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)], 15. Edizioni della Normale, Pisa, 2017. Google Scholar |
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S. Dipierro, G. Palatucci and E. Valdinoci,
Existence and symmetry results for a Schrödinger type problem involving the fractional laplacian, Matematiche (Catania), 68 (2013), 201-216.
|
[19] |
G. Duvaut and J. L. Lions, Inequalities in Mechanics and Physics, Grundlehren Math. Wiss., vol. 219, Springer-Verlag, Berlin, 1976. |
[20] |
P. Felmer, A. Quaas and J. Tan,
Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1237-1262.
doi: 10.1017/S0308210511000746. |
[21] |
G. M. Figueiredo, G. Molica Bisci and R. Servadei,
On a fractional Kirchhoff-type equation via Krasnoselskii's genus, Asymptot. Anal., 94 (2015), 347-361.
doi: 10.3233/ASY-151316. |
[22] |
A. Fiscella and E. Valdinoci,
A critical Kirchhoff type problem involving a nonlocal operator, Nonlinear Anal., 94 (2014), 156-170.
doi: 10.1016/j.na.2013.08.011. |
[23] |
Z. Guo,
Ground states for Kirchhoff equations without compact condition, J. Differential Equations, 259 (2015), 2884-2902.
doi: 10.1016/j.jde.2015.04.005. |
[24] |
X. He and W. Zou,
Ground states for nonlinear Kirchhoff equations with critical growth, Ann. Mat. Pura Appl., 193 (2014), 473-500.
doi: 10.1007/s10231-012-0286-6. |
[25] |
X. He and W. Zou,
Infinitely many positive solutions for Kirchhoff-type problems, Nonlinear Anal., 70 (2009), 1407-1414.
doi: 10.1016/j.na.2008.02.021. |
[26] |
X. He and W. Zou,
Existence and concentration behavior of positive solutions for a Kirchhoff equation in $ \mathbb{R} ^3$, . Differential Equations, 252 (2012), 1813-1834.
doi: 10.1016/j.jde.2011.08.035. |
[27] |
Y. He and G. Li,
Standing waves for a class of Kirchhoff type problems in $ \mathbb{R} ^3$ involving critical Sobolev exponents, Calc. Var. Partial Differential Equations, 54 (2015), 3067-3106.
doi: 10.1007/s00526-015-0894-2. |
[28] |
Y. He, G. Li and S. Peng,
Concentrating bound states for Kirchhoff type problems in $ \mathbb{R} ^3$ involving critical Sobolev exponents, Adv. Nonlinear Stud., 14 (2014), 483-510.
doi: 10.1515/ans-2014-0214. |
[29] |
L. Jeanjean,
On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on $ \mathbb{R} ^N$, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 787-809.
doi: 10.1017/S0308210500013147. |
[30] |
G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883. Google Scholar |
[31] |
N. Laskin,
Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.
doi: 10.1016/S0375-9601(00)00201-2. |
[32] |
N. Laskin,
Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), 56-108.
doi: 10.1103/PhysRevE.66.056108. |
[33] |
G. Li and H. Ye,
Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in $ \mathbb{R} ^3$, J. Differential Equations, 257 (2014), 566-600.
doi: 10.1016/j.jde.2014.04.011. |
[34] |
Z. Liu, M. Squassina and J. Zhang, Ground states for fractional Kirchhoff equations with critical nonlinearity in low dimension, NoDEA Nonlinear Differential Equations Appl., 24 (2017), Art. 50, 32 pp.
doi: 10.1007/s00030-017-0473-7. |
[35] |
E. Milakis and L. Silvestre,
Regularity for the nonlinear Signorini problem, Adv. Math., 217 (2008), 1301-1312.
doi: 10.1016/j.aim.2007.08.009. |
[36] |
P. Pucci and S. Saldi,
Critical stationary Kirchhoff equations in $ \mathbb{R} ^N$ involving nonlocal operators, Rev. Mat. Iberoam., 32 (2016), 1-22.
doi: 10.4171/RMI/879. |
[37] |
P. Pucci, M. Xiang and B. Zhang,
Multiple solutions for nonhomogeneous Schrödinger-Kirchhoff type equations involving the fractional p-Laplacian in $ \mathbb{R} ^N$, Calc. Var. Partial Differential Equations, 54 (2015), 2785-2806.
doi: 10.1007/s00526-015-0883-5. |
[38] |
S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in $ \mathbb{R} ^N$, J. Math. Phys., 54 (2013), 031501, 17pp.
doi: 10.1063/1.4793990. |
[39] |
R. Servadei and E. Valdinoci,
Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.
doi: 10.3934/dcds.2013.33.2105. |
[40] |
R. Servadei and E. Valdinoci,
The Brezis-Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc., 367 (2015), 67-102.
doi: 10.1090/S0002-9947-2014-05884-4. |
[41] |
X. Shang and J. Zhang,
Ground states for fractional Schrödinger equations with critical growth, Nonlinearity, 27 (2014), 187-207.
doi: 10.1088/0951-7715/27/2/187. |
[42] |
L. Silvestre,
Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.
doi: 10.1002/cpa.20153. |
[43] |
X. Tang and S. Chen,
Ground state solutions of Nehari-Pohozaev type for Schrödinger-Poisson problems with general potentials, Discrete Contin. Dyn. Syst., 37 (2017), 4973-5002.
doi: 10.3934/dcds.2017214. |
[44] |
X. Tang and S. Chen, Ground state solutions of Nehari-Pohozaev type for Kirchhoff-type problems with general potentials, Calc. Var. Partial Differential Equations, 56 (2017), Art. 110, 25 pp.
doi: 10.1007/s00526-017-1214-9. |
[45] |
K. Teng,
Existence of ground state solutions for the nonlinear fractional Schrödinger-Poisson system with critical Sobolev exponent, J. Differential Equations, 261 (2017), 3061-3106.
doi: 10.1016/j.jde.2016.05.022. |
[46] |
J. Wang, L. Tian, J. Xu and F. Zhang,
Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth, J. Differential Equations, 253 (2012), 2314-2351.
doi: 10.1016/j.jde.2012.05.023. |
[47] |
M. Willem, Minimax Theorems, Progr. Nonlinear Differential Equations Appl., vol. 24, Birkhäuser Boston, Inc., Boston, MA, 1996.
doi: 10.1007/978-1-4612-4146-1. |
[48] |
X. Wu,
Existence of nontrivial solutions and high energy solutions for Schrödinger-Kirchhoff-type equations in $ \mathbb{R} ^N$, Nonlinear Anal. Real World Appl., 12 (2011), 1278-1287.
doi: 10.1016/j.nonrwa.2010.09.023. |
show all references
References:
[1] |
C. O. Alves, F. J. S. A. Corrêa and T. F. Ma,
Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. Math. Appl., 49 (2005), 85-93.
doi: 10.1016/j.camwa.2005.01.008. |
[2] |
D. Applebaum,
Lévy processes-from probability to finance quantum groups, Notices Am. Math. Soc., 51 (2004), 1336-1347.
|
[3] |
A. Arosio and S. Panizzi,
On the well-posedness of the Kirchhoff string, Trans. Amer. Math. Soc., 348 (1996), 305-330.
doi: 10.1090/S0002-9947-96-01532-2. |
[4] |
P. Biler, G. Karch and W. A. Woyczyński,
Critical nonlinearity exponent and self-similar asymptotics for Lévy conservation laws, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18 (2001), 613-637.
doi: 10.1016/S0294-1449(01)00080-4. |
[5] |
C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, Lecture Notes of the Unione Matematica Italiana, 20. Springer, [Cham]; Unione Matematica Italiana, Bologna, 2016.
doi: 10.1007/978-3-319-28739-3. |
[6] |
L. Caffarelli, J. M. Roquejoffre and O. Savin,
Non-local minimal surfaces, Comm. Pure Appl. Math., 63 (2010), 1111-1144.
doi: 10.1002/cpa.20331. |
[7] |
L. Caffarelli and L. Silvestre,
An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[8] |
L. Caffarelli and E. Valdinoci,
Uniform estimates and limiting arguments for nonlocal minimal surfaces, Calc. Var. Partial Differential Equations, 41 (2011), 203-240.
doi: 10.1007/s00526-010-0359-6. |
[9] |
M. M. Cavalcanti, V. N. Domingos Cavalcanti and J. A. Soriano,
Global existence and uniform decay rates for the Kirchhoff-Carrier equation with nonlinear dissipation, Adv. Differential Equations, 6 (2001), 701-730.
|
[10] |
X. Chang and Z. Wang,
Ground state of scalar field equations involving a fractional Laplacian with general nonlinearity, Nonlinearity, 26 (2013), 479-494.
doi: 10.1088/0951-7715/26/2/479. |
[11] |
C. Chen, Y. Kuo and T. Wu,
The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions, J. Differential Equations, 250 (2011), 1876-1908.
doi: 10.1016/j.jde.2010.11.017. |
[12] |
R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman & Hall/CRC, Boca Raton, FL, 2004. |
[13] |
A. Cotsiolis and N. K. Tavoularis,
Best constants for Sobolev inequalities for higher order fractional derivatives, J. Math. Anal. Appl., 295 (2004), 225-236.
doi: 10.1016/j.jmaa.2004.03.034. |
[14] |
P. D'Ancona and S. Spagnolo,
Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. Math., 108 (1992), 247-262.
doi: 10.1007/BF02100605. |
[15] |
Y. Deng, S. Peng and W. Shuai,
Existence and asympototic behavior of nodal solutions for the Kirchhoff-type problems in $ \mathbb{R} ^3$, J. Funct. Anal., 269 (2015), 3500-3527.
doi: 10.1016/j.jfa.2015.09.012. |
[16] |
E. Di Nezza, G. Palatucci and E. Valdinoci,
Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.
doi: 10.1016/j.bulsci.2011.12.004. |
[17] |
S. Dipierro, M. Medina and E. Valdinoci, Fractional Elliptic Problems with Critical Growth in the Whole of $ \mathbb{R} ^n$, Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)], 15. Edizioni della Normale, Pisa, 2017. Google Scholar |
[18] |
S. Dipierro, G. Palatucci and E. Valdinoci,
Existence and symmetry results for a Schrödinger type problem involving the fractional laplacian, Matematiche (Catania), 68 (2013), 201-216.
|
[19] |
G. Duvaut and J. L. Lions, Inequalities in Mechanics and Physics, Grundlehren Math. Wiss., vol. 219, Springer-Verlag, Berlin, 1976. |
[20] |
P. Felmer, A. Quaas and J. Tan,
Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1237-1262.
doi: 10.1017/S0308210511000746. |
[21] |
G. M. Figueiredo, G. Molica Bisci and R. Servadei,
On a fractional Kirchhoff-type equation via Krasnoselskii's genus, Asymptot. Anal., 94 (2015), 347-361.
doi: 10.3233/ASY-151316. |
[22] |
A. Fiscella and E. Valdinoci,
A critical Kirchhoff type problem involving a nonlocal operator, Nonlinear Anal., 94 (2014), 156-170.
doi: 10.1016/j.na.2013.08.011. |
[23] |
Z. Guo,
Ground states for Kirchhoff equations without compact condition, J. Differential Equations, 259 (2015), 2884-2902.
doi: 10.1016/j.jde.2015.04.005. |
[24] |
X. He and W. Zou,
Ground states for nonlinear Kirchhoff equations with critical growth, Ann. Mat. Pura Appl., 193 (2014), 473-500.
doi: 10.1007/s10231-012-0286-6. |
[25] |
X. He and W. Zou,
Infinitely many positive solutions for Kirchhoff-type problems, Nonlinear Anal., 70 (2009), 1407-1414.
doi: 10.1016/j.na.2008.02.021. |
[26] |
X. He and W. Zou,
Existence and concentration behavior of positive solutions for a Kirchhoff equation in $ \mathbb{R} ^3$, . Differential Equations, 252 (2012), 1813-1834.
doi: 10.1016/j.jde.2011.08.035. |
[27] |
Y. He and G. Li,
Standing waves for a class of Kirchhoff type problems in $ \mathbb{R} ^3$ involving critical Sobolev exponents, Calc. Var. Partial Differential Equations, 54 (2015), 3067-3106.
doi: 10.1007/s00526-015-0894-2. |
[28] |
Y. He, G. Li and S. Peng,
Concentrating bound states for Kirchhoff type problems in $ \mathbb{R} ^3$ involving critical Sobolev exponents, Adv. Nonlinear Stud., 14 (2014), 483-510.
doi: 10.1515/ans-2014-0214. |
[29] |
L. Jeanjean,
On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on $ \mathbb{R} ^N$, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 787-809.
doi: 10.1017/S0308210500013147. |
[30] |
G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883. Google Scholar |
[31] |
N. Laskin,
Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.
doi: 10.1016/S0375-9601(00)00201-2. |
[32] |
N. Laskin,
Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), 56-108.
doi: 10.1103/PhysRevE.66.056108. |
[33] |
G. Li and H. Ye,
Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in $ \mathbb{R} ^3$, J. Differential Equations, 257 (2014), 566-600.
doi: 10.1016/j.jde.2014.04.011. |
[34] |
Z. Liu, M. Squassina and J. Zhang, Ground states for fractional Kirchhoff equations with critical nonlinearity in low dimension, NoDEA Nonlinear Differential Equations Appl., 24 (2017), Art. 50, 32 pp.
doi: 10.1007/s00030-017-0473-7. |
[35] |
E. Milakis and L. Silvestre,
Regularity for the nonlinear Signorini problem, Adv. Math., 217 (2008), 1301-1312.
doi: 10.1016/j.aim.2007.08.009. |
[36] |
P. Pucci and S. Saldi,
Critical stationary Kirchhoff equations in $ \mathbb{R} ^N$ involving nonlocal operators, Rev. Mat. Iberoam., 32 (2016), 1-22.
doi: 10.4171/RMI/879. |
[37] |
P. Pucci, M. Xiang and B. Zhang,
Multiple solutions for nonhomogeneous Schrödinger-Kirchhoff type equations involving the fractional p-Laplacian in $ \mathbb{R} ^N$, Calc. Var. Partial Differential Equations, 54 (2015), 2785-2806.
doi: 10.1007/s00526-015-0883-5. |
[38] |
S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in $ \mathbb{R} ^N$, J. Math. Phys., 54 (2013), 031501, 17pp.
doi: 10.1063/1.4793990. |
[39] |
R. Servadei and E. Valdinoci,
Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.
doi: 10.3934/dcds.2013.33.2105. |
[40] |
R. Servadei and E. Valdinoci,
The Brezis-Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc., 367 (2015), 67-102.
doi: 10.1090/S0002-9947-2014-05884-4. |
[41] |
X. Shang and J. Zhang,
Ground states for fractional Schrödinger equations with critical growth, Nonlinearity, 27 (2014), 187-207.
doi: 10.1088/0951-7715/27/2/187. |
[42] |
L. Silvestre,
Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.
doi: 10.1002/cpa.20153. |
[43] |
X. Tang and S. Chen,
Ground state solutions of Nehari-Pohozaev type for Schrödinger-Poisson problems with general potentials, Discrete Contin. Dyn. Syst., 37 (2017), 4973-5002.
doi: 10.3934/dcds.2017214. |
[44] |
X. Tang and S. Chen, Ground state solutions of Nehari-Pohozaev type for Kirchhoff-type problems with general potentials, Calc. Var. Partial Differential Equations, 56 (2017), Art. 110, 25 pp.
doi: 10.1007/s00526-017-1214-9. |
[45] |
K. Teng,
Existence of ground state solutions for the nonlinear fractional Schrödinger-Poisson system with critical Sobolev exponent, J. Differential Equations, 261 (2017), 3061-3106.
doi: 10.1016/j.jde.2016.05.022. |
[46] |
J. Wang, L. Tian, J. Xu and F. Zhang,
Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth, J. Differential Equations, 253 (2012), 2314-2351.
doi: 10.1016/j.jde.2012.05.023. |
[47] |
M. Willem, Minimax Theorems, Progr. Nonlinear Differential Equations Appl., vol. 24, Birkhäuser Boston, Inc., Boston, MA, 1996.
doi: 10.1007/978-1-4612-4146-1. |
[48] |
X. Wu,
Existence of nontrivial solutions and high energy solutions for Schrödinger-Kirchhoff-type equations in $ \mathbb{R} ^N$, Nonlinear Anal. Real World Appl., 12 (2011), 1278-1287.
doi: 10.1016/j.nonrwa.2010.09.023. |
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