December  2019, 39(12): 6995-7012. doi: 10.3934/dcds.2019241

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

* Corresponding author: Paul H. Rabinowitz

To Luis for his 70th birthday

Received  October 2018 Revised  March 2019 Published  June 2019

Combining situations originally considered in [7] - [8], a semilinear elliptic system is treated and a nondegeneracy condition leading to the existence of multibump solutions is considerably weakened.

Citation: Piero Montecchiari, Paul H. Rabinowitz. A nondegeneracy condition for a semilinear elliptic system and the existence of 1- bump solutions. Discrete & Continuous Dynamical Systems, 2019, 39 (12) : 6995-7012. doi: 10.3934/dcds.2019241
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.  Google Scholar

[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.  Google Scholar

[3]

J. ByeonP. Montecchiari and P. H. Rabinowitz, A double well potential System, Analysis & PDE, 9 (2016), 1737-1772.  doi: 10.2140/apde.2016.9.1737.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

U. Kirchgraber and D. Stoffer, Chaotic behaviour in simple dynamical systems, SIAM Review, 32 (1990), 424-452.  doi: 10.1137/1032078.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[13]

P. MontecchiariM. 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.  Google Scholar

[14]

P. MontecchiariM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

P. H. Rabinowitz, A multibump construction in a degenerate setting, Calc. Var. Partial Differential Equations, 5, 15–182 (1997). doi: 10.1007/s005260050064.  Google Scholar

[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.  Google Scholar

[23]

E. Séré, Existence of infinitely many homoclinic orbits in Hamiltonian systems, Math. Z., 209, 27–42, (1992). doi: 10.1007/BF02570817.  Google Scholar

[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.  Google Scholar

[25]

G. T. Whyburn, Topological Analysis, (Chapter 1), Princeton Univ. Press, Princeton, N. J., (1958).  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

J. ByeonP. Montecchiari and P. H. Rabinowitz, A double well potential System, Analysis & PDE, 9 (2016), 1737-1772.  doi: 10.2140/apde.2016.9.1737.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

U. Kirchgraber and D. Stoffer, Chaotic behaviour in simple dynamical systems, SIAM Review, 32 (1990), 424-452.  doi: 10.1137/1032078.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[13]

P. MontecchiariM. 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.  Google Scholar

[14]

P. MontecchiariM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

P. H. Rabinowitz, A multibump construction in a degenerate setting, Calc. Var. Partial Differential Equations, 5, 15–182 (1997). doi: 10.1007/s005260050064.  Google Scholar

[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.  Google Scholar

[23]

E. Séré, Existence of infinitely many homoclinic orbits in Hamiltonian systems, Math. Z., 209, 27–42, (1992). doi: 10.1007/BF02570817.  Google Scholar

[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.  Google Scholar

[25]

G. T. Whyburn, Topological Analysis, (Chapter 1), Princeton Univ. Press, Princeton, N. J., (1958).  Google Scholar

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