A nondegeneracy condition for a semilinear elliptic system and the existence of 1- bump solutions

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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

Piero Montecchiari ^1, and Paul H. Rabinowitz ^2,,

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.

Keywords: Semilinear elliptic system, nondegeneracy condition, multibump solutions.

Mathematics Subject Classification: Primary: 35J50, 35J47; Secondary: 35J57, 34C37.

Citation: Piero Montecchiari, Paul H. Rabinowitz. A nondegeneracy condition for a semilinear elliptic system and the existence of 1- bump solutions. Discrete and 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.
[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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