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June  2016, 36(6): 3277-3315. doi: 10.3934/dcds.2016.36.3277

Sharp criteria of Liouville type for some nonlinear systems

1. 

Institute of Mathematics, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China

2. 

Department of Applied Mathematical, University of Colorado at Boulder, Boulder, CO 80309, United States

Received  June 2015 Revised  October 2015 Published  December 2015

In this paper, we establish the sharp criteria for the nonexistence of positive solutions to the Hardy-Littlewood-Sobolev (HLS) system of nonlinear equations and the corresponding nonlinear differential systems of Lane-Emden type. These nonexistence results, known as Liouville theorems, are fundamental in PDE theory and applications. A special iteration scheme, a new shooting method and some Pohozaev identities in integral form as well as in differential form are created. Combining these new techniques with some observations and some critical asymptotic analysis, we establish the sharp criteria of Liouville type for our systems of nonlinear equations. Similar results are also derived for the system of Wolff type of integral equations and the system of $\gamma$-Laplace equations. A dichotomy description in terms of existence and nonexistence for solutions with finite energy is also obtained.
Citation: Yutian Lei, Congming Li. Sharp criteria of Liouville type for some nonlinear systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3277-3315. doi: 10.3934/dcds.2016.36.3277
References:
[1]

L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth,, Comm. Pure Appl. Math., 42 (1989), 271.  doi: 10.1002/cpa.3160420304.  Google Scholar

[2]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian,, Comm. Partial Differential Equations, 32 (2007), 1245.  doi: 10.1080/03605300600987306.  Google Scholar

[3]

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.  doi: 10.1007/s00032-008-0090-3.  Google Scholar

[4]

A. Chang and P. Yang, On uniqueness of an n-th order differential equation in conformal geometry,, Math. Res. Lett., 4 (1997), 91.  doi: 10.4310/MRL.1997.v4.n1.a9.  Google Scholar

[5]

W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations,, Duke Math. J., 63 (1991), 615.  doi: 10.1215/S0012-7094-91-06325-8.  Google Scholar

[6]

W. Chen and C. Li, A priori estimates for prescribing scalar curvature equations,, Ann. of Math., 145 (1997), 547.  doi: 10.2307/2951844.  Google Scholar

[7]

W. Chen and C. Li, An integral system and the Lane-Emden conjecture,, Discrete Contin. Dyn. Syst., 24 (2009), 1167.  doi: 10.3934/dcds.2009.24.1167.  Google Scholar

[8]

W. Chen and C. Li, Radial symmetry of solutions for some integral systems of Wolff type,, Discrete Contin. Dyn. Syst., 30 (2011), 1083.  doi: 10.3934/dcds.2011.30.1083.  Google Scholar

[9]

W. Chen and C. Li, Super polyharmonic property of solutions for PDE systems and its applications,, Commun. Pure Appl. Anal., 12 (2013), 2497.  doi: 10.3934/cpaa.2013.12.2497.  Google Scholar

[10]

W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation,, Discrete Contin. Dyn. Syst., 12 (2005), 347.  doi: 10.3934/dcds.2005.12.347.  Google Scholar

[11]

W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations,, Comm. Partial Differential Equations, 30 (2005), 59.  doi: 10.1081/PDE-200044445.  Google Scholar

[12]

W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation,, Commun. Pure Appl. Math., 59 (2006), 330.  doi: 10.1002/cpa.20116.  Google Scholar

[13]

M. Franca, Classification of positive solutions of p-Laplace equation with a growth term,, Arch. Math. (Brno), 40 (2004), 415.   Google Scholar

[14]

F. Gazzola, Critical exponents which relate embedding inequalities with quasilinear elliptic operator,, in Proceedings of the Fourth International Conference on Dynamical Systems and Differential Equations, (2002), 24.   Google Scholar

[15]

F. Gazzola and H.-C. Grunau, Radial entire solutions for supercritical biharmonic equations,, Math. Ann., 334 (2006), 905.  doi: 10.1007/s00208-005-0748-x.  Google Scholar

[16]

B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n$,, in Mathematical Analysis and Applications, (1981), 369.   Google Scholar

[17]

B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations,, Comm. Pure Appl. Math., 34 (1981), 525.  doi: 10.1002/cpa.3160340406.  Google Scholar

[18]

M. Guedda and L. Veron, Local and global properties of solutions of quasilinear elliptic equations,, J. Differential Equations, 76 (1988), 159.  doi: 10.1016/0022-0396(88)90068-X.  Google Scholar

[19]

C. Gui, On positive entire solutions of the elliptic equation $\Delta u+K(x)u^p=0$ and its applications to Riemannian geometry,, Proc. Roy. Soc. Edinburgh Sect. A, 126 (1996), 225.  doi: 10.1017/S0308210500022708.  Google Scholar

[20]

F. Hang, On the integral systems related to Hardy-Littlewood-sobolev inequality,, Math. Res. Lett., 14 (2007), 373.  doi: 10.4310/MRL.2007.v14.n3.a2.  Google Scholar

[21]

L. I. Hedberg and T. Wolff, Thin sets in nonlinear potential theory,, Ann. Inst. Fourier (Grenobel), 33 (1983), 161.  doi: 10.5802/aif.944.  Google Scholar

[22]

C. Jin and C. Li, Qualitative analysis of some systems of integral equations,, Calc. Var. Partial Differential Equations, 26 (2006), 447.  doi: 10.1007/s00526-006-0013-5.  Google Scholar

[23]

N. Kawano, E. Yanagida and S. Yotsutani, Structure theorems for positive radial solutions to div$(|Du|^{m-2} Du)+K(|x|)u^q=0$ in $R^n$,, J. Math. Soc. Japan, 45 (1993), 719.  doi: 10.2969/jmsj/04540719.  Google Scholar

[24]

T. Kilpelaiinen and J. Malý, The Wiener test and potential estimates for quasilinear elliptic equations,, Acta Math., 172 (1994), 137.  doi: 10.1007/BF02392793.  Google Scholar

[25]

D. Labutin, Potential estimates for a class of fully nonlinear elliptic equations,, Duke Math. J., 111 (2002), 1.  doi: 10.1215/S0012-7094-02-11111-9.  Google Scholar

[26]

Y. Lei and C. Li, Integrability and asymptotics of positive solutions of a $\gamma$-Laplace system,, J. Differential Equations, 252 (2012), 2739.  doi: 10.1016/j.jde.2011.10.009.  Google Scholar

[27]

Y. Lei, C. Li and C. Ma, Decay estimation for positive solutions of a $\gamma$-Laplace equation,, Discrete Contin. Dyn. Syst., 30 (2011), 547.  doi: 10.3934/dcds.2011.30.547.  Google Scholar

[28]

Y. Lei, C. Li and C. Ma, Asymptotic radial symmetry and growth estimates of positive solutions to weighted Hardy-Littlewood-Sobolev system,, Calc. Var. Partial Differential Equations, 45 (2012), 43.  doi: 10.1007/s00526-011-0450-7.  Google Scholar

[29]

C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations,, Invent. Math., 123 (1996), 221.  doi: 10.1007/s002220050023.  Google Scholar

[30]

C. Li, A degree theory approach for the shooting method,, , (2013).   Google Scholar

[31]

C. Li and L. Ma, Uniqueness of positive bound states to Schrödinger systems with critical exponents,, SIAM J. Math. Anal., 40 (2008), 1049.  doi: 10.1137/080712301.  Google Scholar

[32]

Y.-Y. Li, Remark on some conformally invariant integral equations: The method of moving spheres,, J. Eur. Math. Soc., 6 (2004), 153.   Google Scholar

[33]

Y. Li, Asymptotic behavior of positive solutions of equation $\Delta u+K(x)u^p=0$ in $R^n$,, J. Differential Equations, 95 (1992), 304.  doi: 10.1016/0022-0396(92)90034-K.  Google Scholar

[34]

Y. Li and W.-M. Ni, On conformal scalar curvature equations in $R^n$,, Duke Math. J., 57 (1988), 895.  doi: 10.1215/S0012-7094-88-05740-7.  Google Scholar

[35]

E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities,, Ann. of Math., 118 (1983), 349.  doi: 10.2307/2007032.  Google Scholar

[36]

C. Lin, A classification of solutions of a conformally invariant fourth order equation in $R^n$,, Comm. Math. Helv., 73 (1998), 206.  doi: 10.1007/s000140050052.  Google Scholar

[37]

J. Liu, Y. Guo and Y. Zhang, Liouville-type theorems for polyharmonic systems in $R^n$,, J. Differential Equations, 225 (2006), 685.  doi: 10.1016/j.jde.2005.10.016.  Google Scholar

[38]

J. Liu, Y. Guo and Y. Zhang, Existence of positive entire solutions for polyharmonic equations and systems,, J. Partial Differential Equations, 19 (2006), 256.   Google Scholar

[39]

C. Ma, W. Chen and C. Li, Regularity of solutions for an integral system of Wolff type,, Adv. Math., 226 (2011), 2676.  doi: 10.1016/j.aim.2010.07.020.  Google Scholar

[40]

E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $R^n$,, Differential Integral Equations, 9 (1996), 465.   Google Scholar

[41]

W.-M. Ni and J. Serrin, Existence and nonexistence theorems for ground states of quasilinear partial differential equations. The anomalous case,, Accad. Naz. Lincei., 77 (1986), 231.   Google Scholar

[42]

L. Nirenberg, Topics in Nonlinear Functional Analysis,, Notes by R. A. Artino, (1974).   Google Scholar

[43]

M. Otani, Existence and nonexistence of nontrivial solutions of some nonlinear degenerate elliptic equations,, J. Funct. Anal., 76 (1988), 140.  doi: 10.1016/0022-1236(88)90053-5.  Google Scholar

[44]

L. A. Peletier and J. Serrin, Ground states for the prescribed mean curvature equation,, Proc. Amer. Math. Soc., 100 (1987), 694.  doi: 10.1090/S0002-9939-1987-0894440-8.  Google Scholar

[45]

N. Phuc and I. Verbitsky, Quasilinear and Hessian equations of Lane-Emden type,, Ann. of Math., 168 (2008), 859.  doi: 10.4007/annals.2008.168.859.  Google Scholar

[46]

P. Pucci and J. Serrin, A general variational identity,, Indiana Univ. Math. J., 35 (1986), 681.  doi: 10.1512/iumj.1986.35.35036.  Google Scholar

[47]

P. Quittner and Ph. Souplet, Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States,, Birkhauser Verlag, (2007).   Google Scholar

[48]

J. Serrin and H. Zou, Non-existence of positive solution of Lane-Emden systems,, Differential Integral Equations, 9 (1996), 635.   Google Scholar

[49]

J. Serrin and H. Zou, Existence of positive solutions of the Lane-Emden system,, Atti Sem. Mat. Fis. Univ. Modena, 46 (1998), 369.   Google Scholar

[50]

J. Serrin and H. Zou, Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities,, Acta Math., 189 (2002), 79.  doi: 10.1007/BF02392645.  Google Scholar

[51]

Ph. Souplet, The proof of the Lane-Emden conjecture in 4 space dimensions,, Adv. Math., 221 (2009), 1409.  doi: 10.1016/j.aim.2009.02.014.  Google Scholar

[52]

S. Sun and Y. Lei, Fast decay estimates for integrable solutions of the Lane-Emden type integral systems involving the Wolff potentials,, J. Funct. Anal., 263 (2012), 3857.  doi: 10.1016/j.jfa.2012.09.012.  Google Scholar

[53]

J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations,, Math. Ann., 313 (1999), 207.  doi: 10.1007/s002080050258.  Google Scholar

[54]

X. Yu, Liouville type theorems for integral equations and integral systems,, Calc. Var. Partial Differential Equations, 46 (2013), 75.  doi: 10.1007/s00526-011-0474-z.  Google Scholar

show all references

References:
[1]

L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth,, Comm. Pure Appl. Math., 42 (1989), 271.  doi: 10.1002/cpa.3160420304.  Google Scholar

[2]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian,, Comm. Partial Differential Equations, 32 (2007), 1245.  doi: 10.1080/03605300600987306.  Google Scholar

[3]

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.  doi: 10.1007/s00032-008-0090-3.  Google Scholar

[4]

A. Chang and P. Yang, On uniqueness of an n-th order differential equation in conformal geometry,, Math. Res. Lett., 4 (1997), 91.  doi: 10.4310/MRL.1997.v4.n1.a9.  Google Scholar

[5]

W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations,, Duke Math. J., 63 (1991), 615.  doi: 10.1215/S0012-7094-91-06325-8.  Google Scholar

[6]

W. Chen and C. Li, A priori estimates for prescribing scalar curvature equations,, Ann. of Math., 145 (1997), 547.  doi: 10.2307/2951844.  Google Scholar

[7]

W. Chen and C. Li, An integral system and the Lane-Emden conjecture,, Discrete Contin. Dyn. Syst., 24 (2009), 1167.  doi: 10.3934/dcds.2009.24.1167.  Google Scholar

[8]

W. Chen and C. Li, Radial symmetry of solutions for some integral systems of Wolff type,, Discrete Contin. Dyn. Syst., 30 (2011), 1083.  doi: 10.3934/dcds.2011.30.1083.  Google Scholar

[9]

W. Chen and C. Li, Super polyharmonic property of solutions for PDE systems and its applications,, Commun. Pure Appl. Anal., 12 (2013), 2497.  doi: 10.3934/cpaa.2013.12.2497.  Google Scholar

[10]

W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation,, Discrete Contin. Dyn. Syst., 12 (2005), 347.  doi: 10.3934/dcds.2005.12.347.  Google Scholar

[11]

W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations,, Comm. Partial Differential Equations, 30 (2005), 59.  doi: 10.1081/PDE-200044445.  Google Scholar

[12]

W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation,, Commun. Pure Appl. Math., 59 (2006), 330.  doi: 10.1002/cpa.20116.  Google Scholar

[13]

M. Franca, Classification of positive solutions of p-Laplace equation with a growth term,, Arch. Math. (Brno), 40 (2004), 415.   Google Scholar

[14]

F. Gazzola, Critical exponents which relate embedding inequalities with quasilinear elliptic operator,, in Proceedings of the Fourth International Conference on Dynamical Systems and Differential Equations, (2002), 24.   Google Scholar

[15]

F. Gazzola and H.-C. Grunau, Radial entire solutions for supercritical biharmonic equations,, Math. Ann., 334 (2006), 905.  doi: 10.1007/s00208-005-0748-x.  Google Scholar

[16]

B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n$,, in Mathematical Analysis and Applications, (1981), 369.   Google Scholar

[17]

B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations,, Comm. Pure Appl. Math., 34 (1981), 525.  doi: 10.1002/cpa.3160340406.  Google Scholar

[18]

M. Guedda and L. Veron, Local and global properties of solutions of quasilinear elliptic equations,, J. Differential Equations, 76 (1988), 159.  doi: 10.1016/0022-0396(88)90068-X.  Google Scholar

[19]

C. Gui, On positive entire solutions of the elliptic equation $\Delta u+K(x)u^p=0$ and its applications to Riemannian geometry,, Proc. Roy. Soc. Edinburgh Sect. A, 126 (1996), 225.  doi: 10.1017/S0308210500022708.  Google Scholar

[20]

F. Hang, On the integral systems related to Hardy-Littlewood-sobolev inequality,, Math. Res. Lett., 14 (2007), 373.  doi: 10.4310/MRL.2007.v14.n3.a2.  Google Scholar

[21]

L. I. Hedberg and T. Wolff, Thin sets in nonlinear potential theory,, Ann. Inst. Fourier (Grenobel), 33 (1983), 161.  doi: 10.5802/aif.944.  Google Scholar

[22]

C. Jin and C. Li, Qualitative analysis of some systems of integral equations,, Calc. Var. Partial Differential Equations, 26 (2006), 447.  doi: 10.1007/s00526-006-0013-5.  Google Scholar

[23]

N. Kawano, E. Yanagida and S. Yotsutani, Structure theorems for positive radial solutions to div$(|Du|^{m-2} Du)+K(|x|)u^q=0$ in $R^n$,, J. Math. Soc. Japan, 45 (1993), 719.  doi: 10.2969/jmsj/04540719.  Google Scholar

[24]

T. Kilpelaiinen and J. Malý, The Wiener test and potential estimates for quasilinear elliptic equations,, Acta Math., 172 (1994), 137.  doi: 10.1007/BF02392793.  Google Scholar

[25]

D. Labutin, Potential estimates for a class of fully nonlinear elliptic equations,, Duke Math. J., 111 (2002), 1.  doi: 10.1215/S0012-7094-02-11111-9.  Google Scholar

[26]

Y. Lei and C. Li, Integrability and asymptotics of positive solutions of a $\gamma$-Laplace system,, J. Differential Equations, 252 (2012), 2739.  doi: 10.1016/j.jde.2011.10.009.  Google Scholar

[27]

Y. Lei, C. Li and C. Ma, Decay estimation for positive solutions of a $\gamma$-Laplace equation,, Discrete Contin. Dyn. Syst., 30 (2011), 547.  doi: 10.3934/dcds.2011.30.547.  Google Scholar

[28]

Y. Lei, C. Li and C. Ma, Asymptotic radial symmetry and growth estimates of positive solutions to weighted Hardy-Littlewood-Sobolev system,, Calc. Var. Partial Differential Equations, 45 (2012), 43.  doi: 10.1007/s00526-011-0450-7.  Google Scholar

[29]

C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations,, Invent. Math., 123 (1996), 221.  doi: 10.1007/s002220050023.  Google Scholar

[30]

C. Li, A degree theory approach for the shooting method,, , (2013).   Google Scholar

[31]

C. Li and L. Ma, Uniqueness of positive bound states to Schrödinger systems with critical exponents,, SIAM J. Math. Anal., 40 (2008), 1049.  doi: 10.1137/080712301.  Google Scholar

[32]

Y.-Y. Li, Remark on some conformally invariant integral equations: The method of moving spheres,, J. Eur. Math. Soc., 6 (2004), 153.   Google Scholar

[33]

Y. Li, Asymptotic behavior of positive solutions of equation $\Delta u+K(x)u^p=0$ in $R^n$,, J. Differential Equations, 95 (1992), 304.  doi: 10.1016/0022-0396(92)90034-K.  Google Scholar

[34]

Y. Li and W.-M. Ni, On conformal scalar curvature equations in $R^n$,, Duke Math. J., 57 (1988), 895.  doi: 10.1215/S0012-7094-88-05740-7.  Google Scholar

[35]

E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities,, Ann. of Math., 118 (1983), 349.  doi: 10.2307/2007032.  Google Scholar

[36]

C. Lin, A classification of solutions of a conformally invariant fourth order equation in $R^n$,, Comm. Math. Helv., 73 (1998), 206.  doi: 10.1007/s000140050052.  Google Scholar

[37]

J. Liu, Y. Guo and Y. Zhang, Liouville-type theorems for polyharmonic systems in $R^n$,, J. Differential Equations, 225 (2006), 685.  doi: 10.1016/j.jde.2005.10.016.  Google Scholar

[38]

J. Liu, Y. Guo and Y. Zhang, Existence of positive entire solutions for polyharmonic equations and systems,, J. Partial Differential Equations, 19 (2006), 256.   Google Scholar

[39]

C. Ma, W. Chen and C. Li, Regularity of solutions for an integral system of Wolff type,, Adv. Math., 226 (2011), 2676.  doi: 10.1016/j.aim.2010.07.020.  Google Scholar

[40]

E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $R^n$,, Differential Integral Equations, 9 (1996), 465.   Google Scholar

[41]

W.-M. Ni and J. Serrin, Existence and nonexistence theorems for ground states of quasilinear partial differential equations. The anomalous case,, Accad. Naz. Lincei., 77 (1986), 231.   Google Scholar

[42]

L. Nirenberg, Topics in Nonlinear Functional Analysis,, Notes by R. A. Artino, (1974).   Google Scholar

[43]

M. Otani, Existence and nonexistence of nontrivial solutions of some nonlinear degenerate elliptic equations,, J. Funct. Anal., 76 (1988), 140.  doi: 10.1016/0022-1236(88)90053-5.  Google Scholar

[44]

L. A. Peletier and J. Serrin, Ground states for the prescribed mean curvature equation,, Proc. Amer. Math. Soc., 100 (1987), 694.  doi: 10.1090/S0002-9939-1987-0894440-8.  Google Scholar

[45]

N. Phuc and I. Verbitsky, Quasilinear and Hessian equations of Lane-Emden type,, Ann. of Math., 168 (2008), 859.  doi: 10.4007/annals.2008.168.859.  Google Scholar

[46]

P. Pucci and J. Serrin, A general variational identity,, Indiana Univ. Math. J., 35 (1986), 681.  doi: 10.1512/iumj.1986.35.35036.  Google Scholar

[47]

P. Quittner and Ph. Souplet, Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States,, Birkhauser Verlag, (2007).   Google Scholar

[48]

J. Serrin and H. Zou, Non-existence of positive solution of Lane-Emden systems,, Differential Integral Equations, 9 (1996), 635.   Google Scholar

[49]

J. Serrin and H. Zou, Existence of positive solutions of the Lane-Emden system,, Atti Sem. Mat. Fis. Univ. Modena, 46 (1998), 369.   Google Scholar

[50]

J. Serrin and H. Zou, Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities,, Acta Math., 189 (2002), 79.  doi: 10.1007/BF02392645.  Google Scholar

[51]

Ph. Souplet, The proof of the Lane-Emden conjecture in 4 space dimensions,, Adv. Math., 221 (2009), 1409.  doi: 10.1016/j.aim.2009.02.014.  Google Scholar

[52]

S. Sun and Y. Lei, Fast decay estimates for integrable solutions of the Lane-Emden type integral systems involving the Wolff potentials,, J. Funct. Anal., 263 (2012), 3857.  doi: 10.1016/j.jfa.2012.09.012.  Google Scholar

[53]

J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations,, Math. Ann., 313 (1999), 207.  doi: 10.1007/s002080050258.  Google Scholar

[54]

X. Yu, Liouville type theorems for integral equations and integral systems,, Calc. Var. Partial Differential Equations, 46 (2013), 75.  doi: 10.1007/s00526-011-0474-z.  Google Scholar

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