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The method of energy channels for nonlinear wave equations
A nondegeneracy condition for a semilinear elliptic system and the existence of 1- bump solutions
1. | Dipartimento di Ingegneria Civile, Edile e Architettura, Università Politecnica delle Marche, Via brecce bianche, Ancona, I-60131, Italy |
2. | Department of Mathematics, University of Wisconsin-Madison, Madison, Wisconsin, 53706, USA |
Combining situations originally considered in [
References:
[1] |
S. Alama and Y. Y. Li,
On "multibump" bound states for certain semilinear elliptic equations, Indiana J. Math., 41 (1992), 983-1026.
doi: 10.1512/iumj.1992.41.41048. |
[2] |
U. Bessi,
A variational proof of a Sitnikov-like theorem, Nonlinear Anal., 20 (1993), 1303-1318.
doi: 10.1016/0362-546X(93)90133-D. |
[3] |
J. Byeon, P. Montecchiari and P. H. Rabinowitz,
A double well potential System, Analysis & PDE, 9 (2016), 1737-1772.
doi: 10.2140/apde.2016.9.1737. |
[4] |
B. Buffoni and E. Séré,
A global condition for quasi-random behaviour in a class of conservative systems, Commun. Pure Appl. Math., 49 (1996), 285-305.
doi: 10.1002/(SICI)1097-0312(199603)49:3<285::AID-CPA3>3.0.CO;2-9. |
[5] |
K. Cieliebak and E. Séré,
Pseudoholomorphic curves and the shadowing lemma, Duke Math. J., 99 (1999), 41-73.
doi: 10.1215/S0012-7094-99-09902-7. |
[6] |
P. Caldiroli and P. Montecchiari,
Homoclinic orbits for second order Hamiltonian systems with potential changing sign, Commun. Appl. Nonlinear Anal., 1 (1994), 97-129.
|
[7] |
V. Coti Zelati and P. H. Rabinowitz,
Homoclinic orbits for second order hamiltonian systems possessing superquadratic potentials, J. Amer. Math. Soc., 4 (1991), 693-727.
doi: 10.1090/S0894-0347-1991-1119200-3. |
[8] |
V. Coti Zelati and P. H. Rabinowitz, Homoclinic type solutions for a semilinear elliptic PDE on Rn, Comm. Pure Appl. Math., 45 (1992), 1217–1269.
doi: 10.1002/cpa.3160451002. |
[9] |
V. Coti Zelati and P. H. Rabinowitz, Multibump periodic solutions of a family of Hamiltonian systems, Top. Meth. in Nonlin. Analysis, 4 (1994), 31-57.
doi: 10.12775/TMNA.1994.022. |
[10] |
U. Kirchgraber and D. Stoffer,
Chaotic behaviour in simple dynamical systems, SIAM Review, 32 (1990), 424-452.
doi: 10.1137/1032078. |
[11] |
P. Montecchiari, Existence and multiplicity of homoclinic solutions for a class of asymptotically periodic second order Hamiltonian systems, Ann. Mat. Pura ed App., CLXVIII (1995), 317–354.
doi: 10.1007/BF01759265. |
[12] |
P. Montecchiari,
Multiplicity results for a class of Semilinear Elliptic Equations on $ \mathbb{R}^m$, Rend. Sem. Mat. Univ. Padova, 95 (1996), 1-36.
|
[13] |
P. Montecchiari, M. Nolasco and S. Terracini,
Multiplicity of homoclinics for a class of time recurrent second order Hamiltonian systems, Calc. Var. Partial Differ., 5 (1997), 523-555.
doi: 10.1007/s005260050078. |
[14] |
P. Montecchiari, M. Nolasco and S. Terracini,
A global condition for periodic Duffing-like equations, Trans. Am. Math. Soc., 351 (1999), 3713-3724.
doi: 10.1090/S0002-9947-99-02249-7. |
[15] |
P. Montecchiari and P. H. Rabinowitz,
On the existence of multi-transition solutions for a class of elliptic systems, Ann. Inst. H. Poincaré Anal. Non Linèaire, 33 (2016), 199-219.
doi: 10.1016/j.anihpc.2014.10.001. |
[16] |
P. Montecchiari and P. H. Rabinowitz, Solutions of mountain pass type for double well potential systems, Calc. Var. PDE, 57 (2018), 114.
doi: 10.1007/s00526-018-1400-4. |
[17] |
P. Montecchiari and P. H. Rabinowitz, On global non-degeneracy conditions for chaotic behavior for a class of dynamical systems, in press, Ann. I. H. Poincaré -AN, (2018).
doi: 10.1016/j.anihpc.2018.08.002. |
[18] |
P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics, 65, Amer. Math. Soc., Providence, R.I., 1986.
doi: 10.1090/cbms/065. |
[19] |
P. H. Rabinowitz,
Homoclinic orbits for a class of Hamiltonian systems, Proc. Roy. Soc. Edinburgh, Sect. A, 114 (1990), 33-38.
doi: 10.1017/S0308210500024240. |
[20] |
P. H. Rabinowitz,
Homoclinic and heteroclinic orbits for a class of Hamiltonian systems, Calc. Variations and P. D. E., 1 (1993), 1-36.
doi: 10.1007/BF02163262. |
[21] |
P. H. Rabinowitz, A multibump construction in a degenerate setting, Calc. Var. Partial Differential Equations, 5, 15–182 (1997).
doi: 10.1007/s005260050064. |
[22] |
P. H. Rabinowitz, On a class of reversible elliptic systems, Networks and Heterogeneous Media, 7, 927–939, (2012).
doi: 10.3934/nhm.2012.7.927. |
[23] |
E. Séré, Existence of infinitely many homoclinic orbits in Hamiltonian systems, Math. Z., 209, 27–42, (1992).
doi: 10.1007/BF02570817. |
[24] |
E. Séré, Looking for the Bernoulli shift, Ann. Inst. H. Poincaré. Anal. Non Linéaire, 10, 561-590, (1993).
doi: 10.1016/S0294-1449(16)30205-0. |
[25] |
G. T. Whyburn, Topological Analysis, (Chapter 1), Princeton Univ. Press, Princeton, N. J., (1958). |
show all references
To Luis for his 70th birthday
References:
[1] |
S. Alama and Y. Y. Li,
On "multibump" bound states for certain semilinear elliptic equations, Indiana J. Math., 41 (1992), 983-1026.
doi: 10.1512/iumj.1992.41.41048. |
[2] |
U. Bessi,
A variational proof of a Sitnikov-like theorem, Nonlinear Anal., 20 (1993), 1303-1318.
doi: 10.1016/0362-546X(93)90133-D. |
[3] |
J. Byeon, P. Montecchiari and P. H. Rabinowitz,
A double well potential System, Analysis & PDE, 9 (2016), 1737-1772.
doi: 10.2140/apde.2016.9.1737. |
[4] |
B. Buffoni and E. Séré,
A global condition for quasi-random behaviour in a class of conservative systems, Commun. Pure Appl. Math., 49 (1996), 285-305.
doi: 10.1002/(SICI)1097-0312(199603)49:3<285::AID-CPA3>3.0.CO;2-9. |
[5] |
K. Cieliebak and E. Séré,
Pseudoholomorphic curves and the shadowing lemma, Duke Math. J., 99 (1999), 41-73.
doi: 10.1215/S0012-7094-99-09902-7. |
[6] |
P. Caldiroli and P. Montecchiari,
Homoclinic orbits for second order Hamiltonian systems with potential changing sign, Commun. Appl. Nonlinear Anal., 1 (1994), 97-129.
|
[7] |
V. Coti Zelati and P. H. Rabinowitz,
Homoclinic orbits for second order hamiltonian systems possessing superquadratic potentials, J. Amer. Math. Soc., 4 (1991), 693-727.
doi: 10.1090/S0894-0347-1991-1119200-3. |
[8] |
V. Coti Zelati and P. H. Rabinowitz, Homoclinic type solutions for a semilinear elliptic PDE on Rn, Comm. Pure Appl. Math., 45 (1992), 1217–1269.
doi: 10.1002/cpa.3160451002. |
[9] |
V. Coti Zelati and P. H. Rabinowitz, Multibump periodic solutions of a family of Hamiltonian systems, Top. Meth. in Nonlin. Analysis, 4 (1994), 31-57.
doi: 10.12775/TMNA.1994.022. |
[10] |
U. Kirchgraber and D. Stoffer,
Chaotic behaviour in simple dynamical systems, SIAM Review, 32 (1990), 424-452.
doi: 10.1137/1032078. |
[11] |
P. Montecchiari, Existence and multiplicity of homoclinic solutions for a class of asymptotically periodic second order Hamiltonian systems, Ann. Mat. Pura ed App., CLXVIII (1995), 317–354.
doi: 10.1007/BF01759265. |
[12] |
P. Montecchiari,
Multiplicity results for a class of Semilinear Elliptic Equations on $ \mathbb{R}^m$, Rend. Sem. Mat. Univ. Padova, 95 (1996), 1-36.
|
[13] |
P. Montecchiari, M. Nolasco and S. Terracini,
Multiplicity of homoclinics for a class of time recurrent second order Hamiltonian systems, Calc. Var. Partial Differ., 5 (1997), 523-555.
doi: 10.1007/s005260050078. |
[14] |
P. Montecchiari, M. Nolasco and S. Terracini,
A global condition for periodic Duffing-like equations, Trans. Am. Math. Soc., 351 (1999), 3713-3724.
doi: 10.1090/S0002-9947-99-02249-7. |
[15] |
P. Montecchiari and P. H. Rabinowitz,
On the existence of multi-transition solutions for a class of elliptic systems, Ann. Inst. H. Poincaré Anal. Non Linèaire, 33 (2016), 199-219.
doi: 10.1016/j.anihpc.2014.10.001. |
[16] |
P. Montecchiari and P. H. Rabinowitz, Solutions of mountain pass type for double well potential systems, Calc. Var. PDE, 57 (2018), 114.
doi: 10.1007/s00526-018-1400-4. |
[17] |
P. Montecchiari and P. H. Rabinowitz, On global non-degeneracy conditions for chaotic behavior for a class of dynamical systems, in press, Ann. I. H. Poincaré -AN, (2018).
doi: 10.1016/j.anihpc.2018.08.002. |
[18] |
P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics, 65, Amer. Math. Soc., Providence, R.I., 1986.
doi: 10.1090/cbms/065. |
[19] |
P. H. Rabinowitz,
Homoclinic orbits for a class of Hamiltonian systems, Proc. Roy. Soc. Edinburgh, Sect. A, 114 (1990), 33-38.
doi: 10.1017/S0308210500024240. |
[20] |
P. H. Rabinowitz,
Homoclinic and heteroclinic orbits for a class of Hamiltonian systems, Calc. Variations and P. D. E., 1 (1993), 1-36.
doi: 10.1007/BF02163262. |
[21] |
P. H. Rabinowitz, A multibump construction in a degenerate setting, Calc. Var. Partial Differential Equations, 5, 15–182 (1997).
doi: 10.1007/s005260050064. |
[22] |
P. H. Rabinowitz, On a class of reversible elliptic systems, Networks and Heterogeneous Media, 7, 927–939, (2012).
doi: 10.3934/nhm.2012.7.927. |
[23] |
E. Séré, Existence of infinitely many homoclinic orbits in Hamiltonian systems, Math. Z., 209, 27–42, (1992).
doi: 10.1007/BF02570817. |
[24] |
E. Séré, Looking for the Bernoulli shift, Ann. Inst. H. Poincaré. Anal. Non Linéaire, 10, 561-590, (1993).
doi: 10.1016/S0294-1449(16)30205-0. |
[25] |
G. T. Whyburn, Topological Analysis, (Chapter 1), Princeton Univ. Press, Princeton, N. J., (1958). |
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