American Institute of Mathematical Sciences

Advanced
April  2017, 37(4): 1789-1818. doi: 10.3934/dcds.2017075

On a resonant mean field type equation: A "critical point at Infinity" approach

Mohameden Ahmedou 1,, , Mohamed Ben Ayed 2, and  Marcello Lucia 3,

1. 

Mathematisches Institut der Justus-Liebig-Universität Giessen, Arndtsrasse 2, D-35392 Giessen, Germany
2. 

Université de Sfax, Faculté des Sciences, Département de Mathématiques, Route de Soukra, Sfax, Tunisia
3. 

The City University of New York, CSI, Mathematics Department, 2800 Victory Boulevard, Staten Island New York 10314, USA

* Corresponding author: Mohameden.Ahmedou@math.uni-giessen.de.

Received  February 2016 Revised  November 2016 Published  December 2016

Fund Project: This work was partially supported by a grant from the Simons Foundation (Nr. 210368 to Marcello Lucia) and the grant MTM2014-52402-C3-1-P (Spain)

We consider the following mean field type equations on domains of
$\mathbb R^2$
 under Dirichlet boundary conditions:
$\left\{ \begin{array}{l} - \Delta u = \varrho \frac{{K {e^u}}}{{\int_\Omega {K {e^u}} }}\;\;\;\;\;{\rm{in}}\;\Omega ,\\\;\;\;\;u = 0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\rm{on}}\;\partial \Omega ,\end{array} \right.$
where
$K$
is a smooth positive function and
$\varrho$
is a positive real parameter.
A "critical point theory at Infinity" approach of A. Bahri to the above problem is developed for the resonant case, i.e. when the parameter
$\varrho$
is a multiple of
$8 π$
. Namely, we identify the so-called "critical points at infinity" of the associated variational problem and compute their Morse indices. We then prove some Bahri-Coron type results which can be seen as a generalization of a degree formula in the non-resonant case due to C.C.Chen and C.S.[18].
Keywords: Mean field equation, critical points at infinity, infinite dimensional Morse theory, variational and topological methods.
Mathematics Subject Classification: Primary:35C60, 35J20;Secondary:35Q70.
Citation: Mohameden Ahmedou, Mohamed Ben Ayed, Marcello Lucia. On a resonant mean field type equation: A "critical point at Infinity" approach. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 1789-1818. doi: 10.3934/dcds.2017075
References:
[1]

T. Aubin, Some Nonlinear Problems in Riemannian Geometry, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-13006-3. Google Scholar

[2]

A. Bahri, Critical Points at Infinity in Some Variational Problems, Pitman Research Notes in Mathematics Series, 182. Longman Scientific Technical, Harlow; copublished in the United States with John Wiley Sons, Inc. , New York, 1989. Google Scholar

[3]

A. Bahri and J.-M. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain, Comm. Pure Appl. Math, 41 (1988), 253-294. doi: 10.1002/cpa.3160410302. Google Scholar

[4]

A. Bahri and J.-M. Coron, The scalar-curvature problem on the standard three-dimensional sphere, J. Funct. Anal., 95 (1991), 106-172. doi: 10.1016/0022-1236(91)90026-2. Google Scholar

[5]

A. Bahri and P. Rabinowitz, Periodic solutions of Hamiltonian systems of 3-body type, Ann. Inst. H. Poincaré Anal. Non Linéaire, 8 (1991), 561-649. Google Scholar

[6]

A. Bahri, An invariant for Yamabe-type flows with applications to scalar-curvature problems in high dimension. A celebration of John F. Nash, Jr, Duke Math. J., 81 (1996), 323-466. doi: 10.1215/S0012-7094-96-08116-8. Google Scholar

[7]

D. Bartolucci and C. S. Lin, Existence and Uniqueness of mean field equation on multiply connected domains at the critical parameter, Math. Ann., 359 (2014), 1-44. doi: 10.1007/s00208-013-0990-6. Google Scholar

[8]

D. Bartolucci and F. De Marchis, Supercritical mean field equations on convex domains and the Onsager's statistical description of two dimensional turbulence, Arch. Ration. Mech. Anal., 217 (2015), 525-570. doi: 10.1007/s00205-014-0836-8. Google Scholar

[9]

M. Ben Ayed, Y. Chen, H. Chtioui and M. Hammami, On the prescribed scalar curvature problem on 4-manifolds, Duke Math. J., 84 (1996), 633-677. doi: 10.1215/S0012-7094-96-08420-3. Google Scholar

[10]

M. Ben Ayed and M. Ould Ahmedou, Existence and multiplicity results for a fourth order mean field equation, J. Funct. Anal., 258 (2010), 3165-3194. doi: 10.1016/j.jfa.2010.01.009. Google Scholar

[11]

H. Brézis and F. Merle, Uniform estimates and blow-up behavior for solutions of ions of −Δu = V (x)eu in two dimensions, Comm. Partial Differential Equations, 16 (1991), 1223-1253. doi: 10.1080/03605309108820797. Google Scholar

[12]

E. Caglioti, P.-L. Lions, C. Marchioro and M. Pulvirenti, A special class of stationary flows for two dimensional Euler equations: a statistical mechanics description, Comm. Math. Phys., 143 (1992), 501-525. doi: 10.1007/BF02099262. Google Scholar

[13]

E. Caglioti, P.-L. Lions, C. Marchioro and M. Pulvirenti, A special class of stationary flows for two dimensional Euler equations: a statistical mechanics description, Part Ⅱ, Comm. Math. Phys., 174 (1995), 229-260. doi: 10.1007/BF02099602. Google Scholar

[14]

A. Chang and P. Yang, Prescribing Gaussian curvature on $\mathbb{S}^2$, Acta Math., 159 (1987), 215-259. doi: 10.1007/BF02392560. Google Scholar

[15]

A. Chang and P. Yang, Conformal deformations of metrics on $\mathbb{S}^2$, J. Diff. Geom., 27 (1988), 259-296. Google Scholar

[16]

A. Chang, C. C. Chen and C. S. Lin, Extremal function of a mean field equation in two dimension, in Lectures on Partial Differential Equations, New Stud. Adv. Math, 2 (2003), 61-93. Google Scholar

[17]

C. C. Chen and C. S. Lin, Sharp estimates for solutions of multi-bubbles in compact Riemann surfaces, Comm. Pure Appl. Math., 55 (2002), 728-771. doi: 10.1002/cpa.3014. Google Scholar

[18]

C. C. Chen and C. S. Lin, Topological degree for a mean field equation on Riemann surfaces, Comm. Pure Appl. Math., 56 (2003), 1667-1727. doi: 10.1002/cpa.10107. Google Scholar

[19]

F. De Marchis, Multiplicity result for a scalar field equation on compact surfaces, Comm. PDE, 33 (2008), 2208-2224. doi: 10.1080/03605300802523446. Google Scholar

[20]

F. De Marchis, Generic multiplicity for a scalar field equation on compact surfaces, J. Funct. Anal., 259 (2010), 2165-2192. doi: 10.1016/j.jfa.2010.07.003. Google Scholar

[21]

F. De Marchis, Multiplicity of solutions for a mean field equation on compact surfaces, Boll. Unione Mat. Ital., 4 (2011), 245-257. Google Scholar

[22]

W. Ding, J. Jost, J. Li and G. Wang, Existence results for mean field equations, Ann. Inst. Henri Poincaré Anal. Non Linéaire, 16 (1999), 653-666. doi: 10.1016/S0294-1449(99)80031-6. Google Scholar

[23]

Z. Djadli, Existence result for the mean field problem on Riemann surfaces of all genus, Commun. Contemp. Math., 10 (2008), 205-220. doi: 10.1142/S0219199708002776. Google Scholar

[24]

Z. Djadli and A. Malchiodi, Existence of conformal metrics with constant Q-curvature, Ann. of Math., 168 (2008), 813-858. doi: 10.4007/annals.2008.168.813. Google Scholar

[25]

P. Esposito, M. Grossi and A. Pistoia, On the existence of blowing up solutions for a mean field equation, Ann. Inst. H. Poincaré Anal. Non Lineéaire, 22 (2005), 227-257. doi: 10.1016/j.anihpc.2004.12.001. Google Scholar

[26]

Z. C. Han, Prescribing Gaussian curvature on S2, Duke Math. J., 61 (1990), 679-703. doi: 10.1215/S0012-7094-90-06125-3. Google Scholar

[27]

Z. C. Han, Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev exponent, Ann. Inst. Poincaré Anal. non linéaire, 8 (1991), 159-174. Google Scholar

[28]

M. K. H. Kiessling, Statistical mechanics of classical particles with logarithmic interactions, Comm. Pure Appl. Math., 46 (1993), 27-56. doi: 10.1002/cpa.3160460103. Google Scholar

[29]

Y. Y. Li and I. Shafrir, Blow-up analysis for solutions of $-Δ u=Ve^{u}$ in dimension two, Indiana Univ. Math. J., 43 (1994), 1255-1270. doi: 10.1512/iumj.1994.43.43054. Google Scholar

[30]

Y. Y. Li, Harnack type inequality: The method of moving planes, Comm. Math. Phys., 200 (1999), 421-444. doi: 10.1007/s002200050536. Google Scholar

[31]

M. Lucia, A deformation Lemma with an application to a mean field equation, Topol. Methods Nonlinear Anal., 30 (2007), 113-138. Google Scholar

[32]

A. Malchiodi, Morse theory and a scalar field equation on compact surfaces, Adv. Differential Equations, 13 (2008), 1109-1129. Google Scholar

[33]

A. Malchiodi, Topological Methods for an elliptic equation with exponential nonlinearities, Discrete Contin. Dyn. Syst., 21 (2008), 277-294. doi: 10.3934/dcds.2008.21.277. Google Scholar

[34] J. Milnor, Lectures on the H-Cobordism Theorem, Princeton University Press, Princeton, 1965. Google Scholar
[35] R. Schoen and S. T. Yau, Lectures on Differential Geometry, International Press, 1994. Google Scholar
[36]

M. Struwe and G. Tarantello, On multivortex solutions in Chern-Simons gauge theory, Boll. Unione. Math. Ital. Sez. B Artic. Ric. Mat.(8), 1 (1998), 109-121. Google Scholar

[37]

G. Tarantello, Selfdual Gauge Field Vortices: An Analytic Approach, Progress in Nonlinear differential equations, 72, Birkhäuser Boston, Inc. Boston, MA, 2008. doi: 10.1007/978-0-8176-4608-0. Google Scholar

[38]

Y. Yang, Solitons in Field Theory and Nonlinear Analysis, Springer monographs in Mathematics, Springer Verlag, New York, Inc, 2001. doi: 10.1007/978-1-4757-6548-9. Google Scholar

[39]

L. Zhang, Blow up solutions of some nonlinear elliptic equation involving exponential nonlinearities, Com. Math. Phys., 268 (2006), 105-133. doi: 10.1007/s00220-006-0092-3. Google Scholar

show all references

References:
[1]

T. Aubin, Some Nonlinear Problems in Riemannian Geometry, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-13006-3. Google Scholar

[2]

A. Bahri, Critical Points at Infinity in Some Variational Problems, Pitman Research Notes in Mathematics Series, 182. Longman Scientific Technical, Harlow; copublished in the United States with John Wiley Sons, Inc. , New York, 1989. Google Scholar

[3]

A. Bahri and J.-M. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain, Comm. Pure Appl. Math, 41 (1988), 253-294. doi: 10.1002/cpa.3160410302. Google Scholar

[4]

A. Bahri and J.-M. Coron, The scalar-curvature problem on the standard three-dimensional sphere, J. Funct. Anal., 95 (1991), 106-172. doi: 10.1016/0022-1236(91)90026-2. Google Scholar

[5]

A. Bahri and P. Rabinowitz, Periodic solutions of Hamiltonian systems of 3-body type, Ann. Inst. H. Poincaré Anal. Non Linéaire, 8 (1991), 561-649. Google Scholar

[6]

A. Bahri, An invariant for Yamabe-type flows with applications to scalar-curvature problems in high dimension. A celebration of John F. Nash, Jr, Duke Math. J., 81 (1996), 323-466. doi: 10.1215/S0012-7094-96-08116-8. Google Scholar

[7]

D. Bartolucci and C. S. Lin, Existence and Uniqueness of mean field equation on multiply connected domains at the critical parameter, Math. Ann., 359 (2014), 1-44. doi: 10.1007/s00208-013-0990-6. Google Scholar

[8]

D. Bartolucci and F. De Marchis, Supercritical mean field equations on convex domains and the Onsager's statistical description of two dimensional turbulence, Arch. Ration. Mech. Anal., 217 (2015), 525-570. doi: 10.1007/s00205-014-0836-8. Google Scholar

[9]

M. Ben Ayed, Y. Chen, H. Chtioui and M. Hammami, On the prescribed scalar curvature problem on 4-manifolds, Duke Math. J., 84 (1996), 633-677. doi: 10.1215/S0012-7094-96-08420-3. Google Scholar

[10]

M. Ben Ayed and M. Ould Ahmedou, Existence and multiplicity results for a fourth order mean field equation, J. Funct. Anal., 258 (2010), 3165-3194. doi: 10.1016/j.jfa.2010.01.009. Google Scholar

[11]

H. Brézis and F. Merle, Uniform estimates and blow-up behavior for solutions of ions of −Δu = V (x)eu in two dimensions, Comm. Partial Differential Equations, 16 (1991), 1223-1253. doi: 10.1080/03605309108820797. Google Scholar

[12]

E. Caglioti, P.-L. Lions, C. Marchioro and M. Pulvirenti, A special class of stationary flows for two dimensional Euler equations: a statistical mechanics description, Comm. Math. Phys., 143 (1992), 501-525. doi: 10.1007/BF02099262. Google Scholar

[13]

E. Caglioti, P.-L. Lions, C. Marchioro and M. Pulvirenti, A special class of stationary flows for two dimensional Euler equations: a statistical mechanics description, Part Ⅱ, Comm. Math. Phys., 174 (1995), 229-260. doi: 10.1007/BF02099602. Google Scholar

[14]

A. Chang and P. Yang, Prescribing Gaussian curvature on $\mathbb{S}^2$, Acta Math., 159 (1987), 215-259. doi: 10.1007/BF02392560. Google Scholar

[15]

A. Chang and P. Yang, Conformal deformations of metrics on $\mathbb{S}^2$, J. Diff. Geom., 27 (1988), 259-296. Google Scholar

[16]

A. Chang, C. C. Chen and C. S. Lin, Extremal function of a mean field equation in two dimension, in Lectures on Partial Differential Equations, New Stud. Adv. Math, 2 (2003), 61-93. Google Scholar

[17]

C. C. Chen and C. S. Lin, Sharp estimates for solutions of multi-bubbles in compact Riemann surfaces, Comm. Pure Appl. Math., 55 (2002), 728-771. doi: 10.1002/cpa.3014. Google Scholar

[18]

C. C. Chen and C. S. Lin, Topological degree for a mean field equation on Riemann surfaces, Comm. Pure Appl. Math., 56 (2003), 1667-1727. doi: 10.1002/cpa.10107. Google Scholar

[19]

F. De Marchis, Multiplicity result for a scalar field equation on compact surfaces, Comm. PDE, 33 (2008), 2208-2224. doi: 10.1080/03605300802523446. Google Scholar

[20]

F. De Marchis, Generic multiplicity for a scalar field equation on compact surfaces, J. Funct. Anal., 259 (2010), 2165-2192. doi: 10.1016/j.jfa.2010.07.003. Google Scholar

[21]

F. De Marchis, Multiplicity of solutions for a mean field equation on compact surfaces, Boll. Unione Mat. Ital., 4 (2011), 245-257. Google Scholar

[22]

W. Ding, J. Jost, J. Li and G. Wang, Existence results for mean field equations, Ann. Inst. Henri Poincaré Anal. Non Linéaire, 16 (1999), 653-666. doi: 10.1016/S0294-1449(99)80031-6. Google Scholar

[23]

Z. Djadli, Existence result for the mean field problem on Riemann surfaces of all genus, Commun. Contemp. Math., 10 (2008), 205-220. doi: 10.1142/S0219199708002776. Google Scholar

[24]

Z. Djadli and A. Malchiodi, Existence of conformal metrics with constant Q-curvature, Ann. of Math., 168 (2008), 813-858. doi: 10.4007/annals.2008.168.813. Google Scholar

[25]

P. Esposito, M. Grossi and A. Pistoia, On the existence of blowing up solutions for a mean field equation, Ann. Inst. H. Poincaré Anal. Non Lineéaire, 22 (2005), 227-257. doi: 10.1016/j.anihpc.2004.12.001. Google Scholar

[26]

Z. C. Han, Prescribing Gaussian curvature on S2, Duke Math. J., 61 (1990), 679-703. doi: 10.1215/S0012-7094-90-06125-3. Google Scholar

[27]

Z. C. Han, Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev exponent, Ann. Inst. Poincaré Anal. non linéaire, 8 (1991), 159-174. Google Scholar

[28]

M. K. H. Kiessling, Statistical mechanics of classical particles with logarithmic interactions, Comm. Pure Appl. Math., 46 (1993), 27-56. doi: 10.1002/cpa.3160460103. Google Scholar

[29]

Y. Y. Li and I. Shafrir, Blow-up analysis for solutions of $-Δ u=Ve^{u}$ in dimension two, Indiana Univ. Math. J., 43 (1994), 1255-1270. doi: 10.1512/iumj.1994.43.43054. Google Scholar

[30]

Y. Y. Li, Harnack type inequality: The method of moving planes, Comm. Math. Phys., 200 (1999), 421-444. doi: 10.1007/s002200050536. Google Scholar

[31]

M. Lucia, A deformation Lemma with an application to a mean field equation, Topol. Methods Nonlinear Anal., 30 (2007), 113-138. Google Scholar

[32]

A. Malchiodi, Morse theory and a scalar field equation on compact surfaces, Adv. Differential Equations, 13 (2008), 1109-1129. Google Scholar

[33]

A. Malchiodi, Topological Methods for an elliptic equation with exponential nonlinearities, Discrete Contin. Dyn. Syst., 21 (2008), 277-294. doi: 10.3934/dcds.2008.21.277. Google Scholar

[34] J. Milnor, Lectures on the H-Cobordism Theorem, Princeton University Press, Princeton, 1965. Google Scholar
[35] R. Schoen and S. T. Yau, Lectures on Differential Geometry, International Press, 1994. Google Scholar
[36]

M. Struwe and G. Tarantello, On multivortex solutions in Chern-Simons gauge theory, Boll. Unione. Math. Ital. Sez. B Artic. Ric. Mat.(8), 1 (1998), 109-121. Google Scholar

[37]

G. Tarantello, Selfdual Gauge Field Vortices: An Analytic Approach, Progress in Nonlinear differential equations, 72, Birkhäuser Boston, Inc. Boston, MA, 2008. doi: 10.1007/978-0-8176-4608-0. Google Scholar

[38]

Y. Yang, Solitons in Field Theory and Nonlinear Analysis, Springer monographs in Mathematics, Springer Verlag, New York, Inc, 2001. doi: 10.1007/978-1-4757-6548-9. Google Scholar

[39]

L. Zhang, Blow up solutions of some nonlinear elliptic equation involving exponential nonlinearities, Com. Math. Phys., 268 (2006), 105-133. doi: 10.1007/s00220-006-0092-3. Google Scholar

[1]

Zalman Balanov, Carlos García-Azpeitia, Wieslaw Krawcewicz. On variational and topological methods in nonlinear difference equations. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2813-2844. doi: 10.3934/cpaa.2018133
[2]

Yves Achdou, Manh-Khang Dao, Olivier Ley, Nicoletta Tchou. A class of infinite horizon mean field games on networks. Networks & Heterogeneous Media, 2019, 14 (3) : 537-566. doi: 10.3934/nhm.2019021
[3]

Gabriella Tarantello. Analytical, geometrical and topological aspects of a class of mean field equations on surfaces. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 931-973. doi: 10.3934/dcds.2010.28.931
[4]

Martin Schechter. Monotonicity methods for infinite dimensional sandwich systems. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 455-468. doi: 10.3934/dcds.2010.28.455
[5]

Luis Barreira, Claudia Valls. Topological conjugacies and behavior at infinity. Communications on Pure & Applied Analysis, 2014, 13 (2) : 687-701. doi: 10.3934/cpaa.2014.13.687
[6]

Andrea Malchiodi. Topological methods for an elliptic equation with exponential nonlinearities. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 277-294. doi: 10.3934/dcds.2008.21.277
[7]

Olivier Guéant. New numerical methods for mean field games with quadratic costs. Networks & Heterogeneous Media, 2012, 7 (2) : 315-336. doi: 10.3934/nhm.2012.7.315
[8]

Jaume Llibre, Jesús S. Pérez del Río, J. Angel Rodríguez. Structural stability of planar semi-homogeneous polynomial vector fields applications to critical points and to infinity. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 809-828. doi: 10.3934/dcds.2000.6.809
[9]

Xiaoyu Zheng, Peter Palffy-Muhoray. One order parameter tensor mean field theory for biaxial liquid crystals. Discrete & Continuous Dynamical Systems - B, 2011, 15 (2) : 475-490. doi: 10.3934/dcdsb.2011.15.475
[10]

Philip Schrader. Morse theory for elastica. Journal of Geometric Mechanics, 2016, 8 (2) : 235-256. doi: 10.3934/jgm.2016006
[11]

Futoshi Takahashi. Morse indices and the number of blow up points of blowing-up solutions for a Liouville equation with singular data. Conference Publications, 2013, 2013 (special) : 729-736. doi: 10.3934/proc.2013.2013.729
[12]

Hayato Chiba, Georgi S. Medvedev. The mean field analysis of the Kuramoto model on graphs Ⅰ. The mean field equation and transition point formulas. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 131-155. doi: 10.3934/dcds.2019006
[13]

Diogo Gomes, Levon Nurbekyan. An infinite-dimensional weak KAM theory via random variables. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6167-6185. doi: 10.3934/dcds.2016069
[14]

Silvia Cingolani, Mónica Clapp. Symmetric semiclassical states to a magnetic nonlinear Schrödinger equation via equivariant Morse theory. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1263-1281. doi: 10.3934/cpaa.2010.9.1263
[15]

Qiong Meng, X. H. Tang. Multiple solutions of second-order ordinary differential equation via Morse theory. Communications on Pure & Applied Analysis, 2012, 11 (3) : 945-958. doi: 10.3934/cpaa.2012.11.945
[16]

Jianhui Huang, Xun Li, Jiongmin Yong. A linear-quadratic optimal control problem for mean-field stochastic differential equations in infinite horizon. Mathematical Control & Related Fields, 2015, 5 (1) : 97-139. doi: 10.3934/mcrf.2015.5.97
[17]

Yinbin Deng, Yi Li, Wei Shuai. Existence of solutions for a class of p-Laplacian type equation with critical growth and potential vanishing at infinity. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 683-699. doi: 10.3934/dcds.2016.36.683
[18]

Yonghui Zhou, Jian Yu, Long Wang. Topological essentiality in infinite games. Journal of Industrial & Management Optimization, 2012, 8 (1) : 179-187. doi: 10.3934/jimo.2012.8.179
[19]

Chjan C. Lim. Extremal free energy in a simple mean field theory for a coupled Barotropic fluid - rotating sphere system. Discrete & Continuous Dynamical Systems - A, 2007, 19 (2) : 361-386. doi: 10.3934/dcds.2007.19.361
[20]

J. C. Robinson. A topological time-delay embedding theorem for infinite-dimensional cocycle dynamical systems. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 731-741. doi: 10.3934/dcdsb.2008.9.731

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (16)
  • HTML views (3)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences