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Bifurcation results on positive solutions of an indefinite nonlinear elliptic system
Instability of periodic minimals
1. | Departamento de Matematica Aplicada, Facultad de Ciencias, Universidad de Granada, E-18071, Granada, Spain |
References:
[1] |
S. V. Bolotin and V. V. Kozlov, Asymptotic solutions of the equations of dynamics, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 102 (1980), 84-89. |
[2] |
C. Carathéodory, "Calculus of Variations and Partial Differential Equations ofthe First Order," Chelsea, New York, 1989. |
[3] |
E. N. Dancer and R. Ortega, The index of Lyapunov stable fixed points in two dimensions, J. Dynam. Differential Equations, 6 (1994), 631-637.
doi: 10.1007/BF02218851. |
[4] |
P. Hagedorn, Die Umkehrung der Stabilitätssätze von Lagrange-Dirichlet und Routh, Arch. Rational Mech. Anal., 42 (1971), 281-316.
doi: 10.1007/BF00282334. |
[5] |
P. Hagedorn and J. Mawhin, A simple variational approach to a converse of the Lagrange-Dirichlet theorem, Arch. Rational Mech. Anal., 120 (1992), 327-335.
doi: 10.1007/BF00380318. |
[6] |
J. N. Mather, Variational construction of orbits of twist diffeomorphisms, J. Amer. Math. Soc., 4 (1991), 207-263.
doi: 10.1090/S0894-0347-1991-1080112-5. |
[7] |
J. Moser, "Selected Chapters in the Calculus of Variations," Lecture notes by Oliver Knill. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2003. |
[8] |
R. Ortega, The number of stable periodic solutions of time-dependentHamiltonian systems with one degree of freedom, Ergodic TheoryDynam. Systems, 18 (1998), 1007-1018.
doi: 10.1017/S0143385798108362. |
[9] |
R. Ortega, Instability of periodic solutions obtained by minimization, 'The first 60 years of Nonlinear Analysis of Jean Mawhin', World Scientific, (2004), 189-197. . |
[10] |
P. H. Rabinowitz, Heteroclinics for a reversible Hamiltonian system, Ergodic Theory Dynam. Systems, 14 (1994), 817-829.
doi: 10.1017/S0143385700008178. |
[11] |
P. H. Rabinowitz, A note on a class of reversibleHamiltonian systems, Adv. Nonlinear Stud., 9 (2009), 815-823. |
[12] |
A. J. Ureña, Isolated periodic minima are unstable, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 877-889. |
[13] |
A. J. Ureña, All periodic minimizers are unstable, Arch. Math., 91 (2008), 63-75.
doi: 10.1007/s00013-008-2693-x. |
show all references
References:
[1] |
S. V. Bolotin and V. V. Kozlov, Asymptotic solutions of the equations of dynamics, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 102 (1980), 84-89. |
[2] |
C. Carathéodory, "Calculus of Variations and Partial Differential Equations ofthe First Order," Chelsea, New York, 1989. |
[3] |
E. N. Dancer and R. Ortega, The index of Lyapunov stable fixed points in two dimensions, J. Dynam. Differential Equations, 6 (1994), 631-637.
doi: 10.1007/BF02218851. |
[4] |
P. Hagedorn, Die Umkehrung der Stabilitätssätze von Lagrange-Dirichlet und Routh, Arch. Rational Mech. Anal., 42 (1971), 281-316.
doi: 10.1007/BF00282334. |
[5] |
P. Hagedorn and J. Mawhin, A simple variational approach to a converse of the Lagrange-Dirichlet theorem, Arch. Rational Mech. Anal., 120 (1992), 327-335.
doi: 10.1007/BF00380318. |
[6] |
J. N. Mather, Variational construction of orbits of twist diffeomorphisms, J. Amer. Math. Soc., 4 (1991), 207-263.
doi: 10.1090/S0894-0347-1991-1080112-5. |
[7] |
J. Moser, "Selected Chapters in the Calculus of Variations," Lecture notes by Oliver Knill. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2003. |
[8] |
R. Ortega, The number of stable periodic solutions of time-dependentHamiltonian systems with one degree of freedom, Ergodic TheoryDynam. Systems, 18 (1998), 1007-1018.
doi: 10.1017/S0143385798108362. |
[9] |
R. Ortega, Instability of periodic solutions obtained by minimization, 'The first 60 years of Nonlinear Analysis of Jean Mawhin', World Scientific, (2004), 189-197. . |
[10] |
P. H. Rabinowitz, Heteroclinics for a reversible Hamiltonian system, Ergodic Theory Dynam. Systems, 14 (1994), 817-829.
doi: 10.1017/S0143385700008178. |
[11] |
P. H. Rabinowitz, A note on a class of reversibleHamiltonian systems, Adv. Nonlinear Stud., 9 (2009), 815-823. |
[12] |
A. J. Ureña, Isolated periodic minima are unstable, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 877-889. |
[13] |
A. J. Ureña, All periodic minimizers are unstable, Arch. Math., 91 (2008), 63-75.
doi: 10.1007/s00013-008-2693-x. |
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