November  2019, 12(7): 1929-1954. doi: 10.3934/dcdss.2019126

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

* Corresponding author: Yinbin Deng

Received  November 2017 Revised  April 2018 Published  December 2018

We study the following fractional Kirchhoff type equation:
$ \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*} $
where
$ a, \ b>0 $
are constants,
$ 2^*_s = \frac{6}{3-2s} $
with
$ s\in(0, 1) $
is the critical Sobolev exponent in
$ \mathbb{R} ^3 $
,
$ V $
is a potential function on
$ \mathbb{R} ^3 $
. Under some more general assumptions on
$ f $
and
$ V $
, we prove that the given problem admits a least energy solution by using a constrained minimization on Nehari-Pohozaev manifold and monotone method.
Citation: Yinbin Deng, Wentao Huang. Least energy solutions for fractional Kirchhoff type equations involving critical growth. Discrete and Continuous Dynamical Systems - S, 2019, 12 (7) : 1929-1954. doi: 10.3934/dcdss.2019126
References:
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C. O. AlvesF. 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. BilerG. 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.

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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.

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L. CaffarelliJ. M. Roquejoffre and O. Savin, Non-local minimal surfaces, Comm. Pure Appl. Math., 63 (2010), 1111-1144.  doi: 10.1002/cpa.20331.

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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. CavalcantiV. 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. 

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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.

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C. ChenY. 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.

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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.

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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. DengS. 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.

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E. Di NezzaG. 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.

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S. DipierroG. Palatucci and E. Valdinoci, Existence and symmetry results for a Schrödinger type problem involving the fractional laplacian, Matematiche (Catania), 68 (2013), 201-216. 

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P. FelmerA. 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. FigueiredoG. 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. HeG. 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.

[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. PucciM. 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. WangL. TianJ. 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. AlvesF. 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. BilerG. 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. CaffarelliJ. 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. CavalcantiV. 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. ChenY. 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. DengS. 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 NezzaG. 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.

[18]

S. DipierroG. 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. FelmerA. 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. FigueiredoG. 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. HeG. 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.

[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. PucciM. 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. WangL. TianJ. 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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