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March  2014, 13(2): 623-634. doi: 10.3934/cpaa.2014.13.623

Infinitely many homoclinic solutions for damped vibration problems with subquadratic potentials

Ziheng Zhang 1, and  Rong Yuan 2,

1. 

Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China
2. 

School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875

Received  January 2013 Revised  April 2013 Published  October 2013

In this paper we investigate the existence of infinitely many homoclinic solutions for the following damped vibration problems \begin{eqnarray} \ddot q+A \dot q-L(t)q+W_q(t,q)=0, \end{eqnarray} where $A$ is an antisymmetric constant matrix, $L\in C(R,R^{n^2})$ is a symmetric and positive definite matrix for all $t\in R$, $W\in C^1(R\times R^n,R)$. The novelty of this paper is that, for the case that $W$ is subquadratic at infinity, we establish two new criteria to guarantee the existence of infinitely many homoclinic solutions for (DS) via the genus properties in critical point theory. Recent results in the literature are generalized and significantly improved.
Keywords: critical point, Homoclinic solutions, genus., variational methods.
Mathematics Subject Classification: Primary: 34C37, 35A15; Secondary: 35B3.
Citation: Ziheng Zhang, Rong Yuan. Infinitely many homoclinic solutions for damped vibration problems with subquadratic potentials. Communications on Pure & Applied Analysis, 2014, 13 (2) : 623-634. doi: 10.3934/cpaa.2014.13.623
References:
[1]

C. O. Alves, P. C. Carrião and O. H. Miyagaki, paper title is capitalized., Existence of homoclinic orbits for asymptotically periodic systems involving Duffing-like equation,, Appl. Math. Lett., 16 (2003), 639.  doi: 10.1016/S0893-9659(03)00059-4.  Google Scholar

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[3]

P. Caldiroli and P. Montecchiari, Homoclinic orbits for second order Hamiltonian systems with potential changing sign,, Comm. Appl. Nonlinear Anal., 1 (1994), 97.   Google Scholar

[4]

V. Coti Zelati and P. H. Rabinowitz, Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials,, J. Amer. Math. Soc., 4 (1991), 693.  doi: 10.1090/S0894-0347-1991-1119200-3.  Google Scholar

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A. Daouas, Homoclinic solutions for superquadratic Hamiltonian systems without periodicity assumption,, Nonlinear Anal., 74 (2011), 3407.  doi: 10.1016/j.na.2011.03.001.  Google Scholar

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Y. Ding, Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems,, Nonlinear Anal., 25 (1995), 1095.  doi: 10.1016/0362-546X(94)00229-B.  Google Scholar

[7]

Y. Ding and C. Lee, Homoclinics for asymptotically quadratic and supequadratic Hamiltonian systems,, Nonlinear Anal., 71 (2009), 1395.  doi: 10.1016/j.na.2008.10.116.  Google Scholar

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M. Izydorek and J. Janczewska, Homoclinic solutions for a class of the second order Hamiltonian systems,, J. Differential Equations, 219 (2005), 375.  doi: 10.1016/j.jde.2005.06.029.  Google Scholar

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X. Lv and J. Jiang, Existence of homoclinic solutions for a class of second-order Hamiltonian systems with general potentials,, Nonlinear Analysis: Real World Applications, 13 (2012), 1152.  doi: 10.1016/j.nonrwa.2011.09.008.  Google Scholar

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W. Omana and M. Willem, Homoclinic orbits for a class of Hamiltonian systems,, Differential Integral Equations, 5 (1992), 1115.   Google Scholar

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H. Poincaré, Les méthodes nouvelles de la mécanique céleste,, Gauthier-Villars, (): 1897.   Google Scholar

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P. H. Rabinowitz, "Minimax Methods in Critical Point Theory with Applications to Differential Equations,", CBMS Reg. Conf. Ser. in. Math., (1986).   Google Scholar

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P. H. Rabinowitz, Homoclinic orbits for a class of Hamiltonian systems,, Proc. Roy. Soc. Edinburgh Sect. A., 114 (1990), 33.  doi: 10.1017/S0308210500024240.  Google Scholar

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P. H. Rabinowitz and K. Tanaka, Some results on connecting orbits for a class of Hamiltonian systems,, Math. Z., 206 (1991), 473.  doi: 10.1007/BF02571356.  Google Scholar

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A. Salvatore, Homoclinic orbits for a special class of nonautonomous Hamiltonian systems,, Proceedings of the Second World Congress of Nonlinear Analysis, 30 (1997), 4849.  doi: SO362-546X(97)00142-9.  Google Scholar

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J. Sun, H. Chen and J. J. Nieto, Homoclinic solutions for a class of subquadratic second-order Hamiltonian systems,, J. Math. Anal. Appl., 373 (2011), 20.  doi: 10.1016/j.jmaa.2010.06.038.  Google Scholar

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J. Sun, J. J. Nieto and M. Otero-Novoa, On homoclinic orbits for a class of damped vibration systems,, Advances in Difference Equations, 2012 ().  doi: 10.1186/1687-1847-2012-102.  Google Scholar

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[22]

X. Tang and X. Lin, Existence of infinitely many homoclinic orbits in Hamiltonian systems,, Proc. Roy. Soc. Edinburgh Sect. A., 141 (2011), 1103.  doi: 10.1017/S0308210509001346.  Google Scholar

[23]

L. Wan and C. Tang, Existence and multiplicity of homoclinic orbits for second order Hamiltonian systems without (AR) condition,, Discrete. Cont. Dyn. Syst. Ser. B, 15 (2011), 255.  doi: 10.3934/dcdsb.2011.15.255.  Google Scholar

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[25]

J. Wang, F. Zhang and J. Xu, Existence and multiplicity of homoclinic orbits for the second order Hamiltonian systems,, J. Math. Anal. Appl., 366 (2010), 569.  doi: 10.1016/j.jmaa.2010.01.060.  Google Scholar

[26]

J. Wei and J, Wang, Infinitely many homoclinic orbits for the second order Hamiltonian systems with general potentials,, J. Math. Anal. Anal., 366 (2010), 694.  doi: 10.1016/j.jmaa.2009.12.024.  Google Scholar

[27]

X. Wu and W. Zhang, Existence and multiplicity of homoclinic solutions for a class of damped vibration problems,, Nonlinear Anal., 74 (2011), 4392.  doi: 10.1016/j.na.2011.03.059.  Google Scholar

[28]

M. Yang and Z. Han, The existence of homoclinic solutions for second-order Hamiltonian systems with periodic potentials,, Nonlinear Analysis: Real World Applications, 12 (2011), 2742.  doi: 10.1016/j.nonrwa.2011.03.019.  Google Scholar

[29]

M. Yang and Z. Han, Infinitely many homoclinic solutions for second-order Hamiltonian systems with odd nonlinearities,, Nonlinear Anal., 74 (2011), 2635.  doi: 10.1016/j.na.2010.12.019.  Google Scholar

[30]

R. Yuan and Z. Zhang, Homoclinic solutions for a class of second order non-autonomous systems,, Electron. J. of Differential Equations, 128 (2009), 1.   Google Scholar

[31]

Q. Zhang and C. Liu, Infinitely many homoclinic solutions for second order Hamiltonian systems,, Nonlinear Anal., 72 (2010), 894.  doi: 10.1016/j.na.2009.07.021.  Google Scholar

[32]

Z. Zhang and R. Yuan, Homoclinic solutions for a class of non-autonomous subquadratic second order Hamiltonian systems,, Nonlinear Anal., 71 (2009), 4125.  doi: 10.1016/j.na.2009.02.071.  Google Scholar

[33]

Z. Zhang and R. Yuan, Homoclinic solutions of some second order non-autonomous systems,, Nonlinear Anal., 71 (2009), 5790.  doi: 10.1016/j.na.2009.05.003.  Google Scholar

[34]

Z. Zhang and R. Yuan, Homoclinic solutions for a class of asymptotically quadratic Hamiltonian systems,, Nonlinear Analysis: Real World Applications, 11 (2010), 4185.  doi: 10.1016/j.nonrwa.2010.05.005.  Google Scholar

[35]

W. Zhu, Existence of homoclinic solutions for a class of second order systems,, Nonlinear Anal., 75 (2012), 2455.  doi: 10.1016/j.na.2011.10.043.  Google Scholar

[36]

W. Zou and S. Li, Infinitely many homoclinic orbits for the second-order Hamiltonian systems,, Appl. Math. Lett., 16 (2003), 1283.  doi: 10.1016/S0893-9659(03)90130-3.  Google Scholar

show all references

References:
[1]

C. O. Alves, P. C. Carrião and O. H. Miyagaki, paper title is capitalized., Existence of homoclinic orbits for asymptotically periodic systems involving Duffing-like equation,, Appl. Math. Lett., 16 (2003), 639.  doi: 10.1016/S0893-9659(03)00059-4.  Google Scholar

[2]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications,, J. Funct. Anal., 14 (1973), 349.  doi: 10.1016/0022-1236(73)90051-7.  Google Scholar

[3]

P. Caldiroli and P. Montecchiari, Homoclinic orbits for second order Hamiltonian systems with potential changing sign,, Comm. Appl. Nonlinear Anal., 1 (1994), 97.   Google Scholar

[4]

V. Coti Zelati and P. H. Rabinowitz, Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials,, J. Amer. Math. Soc., 4 (1991), 693.  doi: 10.1090/S0894-0347-1991-1119200-3.  Google Scholar

[5]

A. Daouas, Homoclinic solutions for superquadratic Hamiltonian systems without periodicity assumption,, Nonlinear Anal., 74 (2011), 3407.  doi: 10.1016/j.na.2011.03.001.  Google Scholar

[6]

Y. Ding, Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems,, Nonlinear Anal., 25 (1995), 1095.  doi: 10.1016/0362-546X(94)00229-B.  Google Scholar

[7]

Y. Ding and C. Lee, Homoclinics for asymptotically quadratic and supequadratic Hamiltonian systems,, Nonlinear Anal., 71 (2009), 1395.  doi: 10.1016/j.na.2008.10.116.  Google Scholar

[8]

M. Izydorek and J. Janczewska, Homoclinic solutions for a class of the second order Hamiltonian systems,, J. Differential Equations, 219 (2005), 375.  doi: 10.1016/j.jde.2005.06.029.  Google Scholar

[9]

P. Korman and A. C. Lazer, Homoclinic orbits for a class of symmetric Hamiltonian systems,, Electron. J. Differential Equations, (1994), 1.  doi: 10.1.1.27.6093.  Google Scholar

[10]

X. Lv and J. Jiang, Existence of homoclinic solutions for a class of second-order Hamiltonian systems with general potentials,, Nonlinear Analysis: Real World Applications, 13 (2012), 1152.  doi: 10.1016/j.nonrwa.2011.09.008.  Google Scholar

[11]

X. Lv, S. Lu and P. Yan, Existence of homoclinic solutions for a class of second-order Hamiltonian systems,, Nonlinear Anal., 72 (2010), 390.  doi: 10.1016/j.na.2009.06.073.  Google Scholar

[12]

Y. Lv and C. Tang, Existence of even homoclinic orbits for a class of Hamiltonian systems,, Nonlinear Anal., 67 (2007), 2189.  doi: 10.1016/j.na.2006.08.043.  Google Scholar

[13]

W. Omana and M. Willem, Homoclinic orbits for a class of Hamiltonian systems,, Differential Integral Equations, 5 (1992), 1115.   Google Scholar

[14]

H. Poincaré, Les méthodes nouvelles de la mécanique céleste,, Gauthier-Villars, (): 1897.   Google Scholar

[15]

P. H. Rabinowitz, "Minimax Methods in Critical Point Theory with Applications to Differential Equations,", CBMS Reg. Conf. Ser. in. Math., (1986).   Google Scholar

[16]

P. H. Rabinowitz, Homoclinic orbits for a class of Hamiltonian systems,, Proc. Roy. Soc. Edinburgh Sect. A., 114 (1990), 33.  doi: 10.1017/S0308210500024240.  Google Scholar

[17]

P. H. Rabinowitz and K. Tanaka, Some results on connecting orbits for a class of Hamiltonian systems,, Math. Z., 206 (1991), 473.  doi: 10.1007/BF02571356.  Google Scholar

[18]

A. Salvatore, Homoclinic orbits for a special class of nonautonomous Hamiltonian systems,, Proceedings of the Second World Congress of Nonlinear Analysis, 30 (1997), 4849.  doi: SO362-546X(97)00142-9.  Google Scholar

[19]

J. Sun, H. Chen and J. J. Nieto, Homoclinic solutions for a class of subquadratic second-order Hamiltonian systems,, J. Math. Anal. Appl., 373 (2011), 20.  doi: 10.1016/j.jmaa.2010.06.038.  Google Scholar

[20]

J. Sun, J. J. Nieto and M. Otero-Novoa, On homoclinic orbits for a class of damped vibration systems,, Advances in Difference Equations, 2012 ().  doi: 10.1186/1687-1847-2012-102.  Google Scholar

[21]

X. Tang and X. Lin, Infinitely many homoclinic orbits for Hamiltonian systems with indefinite subquadratic potentials,, Nonlinear Anal., 74 (2011), 6314.  doi: 10.1016/j.na.2011.06.010.  Google Scholar

[22]

X. Tang and X. Lin, Existence of infinitely many homoclinic orbits in Hamiltonian systems,, Proc. Roy. Soc. Edinburgh Sect. A., 141 (2011), 1103.  doi: 10.1017/S0308210509001346.  Google Scholar

[23]

L. Wan and C. Tang, Existence and multiplicity of homoclinic orbits for second order Hamiltonian systems without (AR) condition,, Discrete. Cont. Dyn. Syst. Ser. B, 15 (2011), 255.  doi: 10.3934/dcdsb.2011.15.255.  Google Scholar

[24]

J. Wang, J. Xu and F. Zhang, Homoclinic orbits for a class of Hamiltonian systems with superquadratic or asymptotically quadratic potentials,, Comm. Pure. Appl. Anal., 10 (2011), 269.  doi: 10.3934/cpaa.2011.10.269.  Google Scholar

[25]

J. Wang, F. Zhang and J. Xu, Existence and multiplicity of homoclinic orbits for the second order Hamiltonian systems,, J. Math. Anal. Appl., 366 (2010), 569.  doi: 10.1016/j.jmaa.2010.01.060.  Google Scholar

[26]

J. Wei and J, Wang, Infinitely many homoclinic orbits for the second order Hamiltonian systems with general potentials,, J. Math. Anal. Anal., 366 (2010), 694.  doi: 10.1016/j.jmaa.2009.12.024.  Google Scholar

[27]

X. Wu and W. Zhang, Existence and multiplicity of homoclinic solutions for a class of damped vibration problems,, Nonlinear Anal., 74 (2011), 4392.  doi: 10.1016/j.na.2011.03.059.  Google Scholar

[28]

M. Yang and Z. Han, The existence of homoclinic solutions for second-order Hamiltonian systems with periodic potentials,, Nonlinear Analysis: Real World Applications, 12 (2011), 2742.  doi: 10.1016/j.nonrwa.2011.03.019.  Google Scholar

[29]

M. Yang and Z. Han, Infinitely many homoclinic solutions for second-order Hamiltonian systems with odd nonlinearities,, Nonlinear Anal., 74 (2011), 2635.  doi: 10.1016/j.na.2010.12.019.  Google Scholar

[30]

R. Yuan and Z. Zhang, Homoclinic solutions for a class of second order non-autonomous systems,, Electron. J. of Differential Equations, 128 (2009), 1.   Google Scholar

[31]

Q. Zhang and C. Liu, Infinitely many homoclinic solutions for second order Hamiltonian systems,, Nonlinear Anal., 72 (2010), 894.  doi: 10.1016/j.na.2009.07.021.  Google Scholar

[32]

Z. Zhang and R. Yuan, Homoclinic solutions for a class of non-autonomous subquadratic second order Hamiltonian systems,, Nonlinear Anal., 71 (2009), 4125.  doi: 10.1016/j.na.2009.02.071.  Google Scholar

[33]

Z. Zhang and R. Yuan, Homoclinic solutions of some second order non-autonomous systems,, Nonlinear Anal., 71 (2009), 5790.  doi: 10.1016/j.na.2009.05.003.  Google Scholar

[34]

Z. Zhang and R. Yuan, Homoclinic solutions for a class of asymptotically quadratic Hamiltonian systems,, Nonlinear Analysis: Real World Applications, 11 (2010), 4185.  doi: 10.1016/j.nonrwa.2010.05.005.  Google Scholar

[35]

W. Zhu, Existence of homoclinic solutions for a class of second order systems,, Nonlinear Anal., 75 (2012), 2455.  doi: 10.1016/j.na.2011.10.043.  Google Scholar

[36]

W. Zou and S. Li, Infinitely many homoclinic orbits for the second-order Hamiltonian systems,, Appl. Math. Lett., 16 (2003), 1283.  doi: 10.1016/S0893-9659(03)90130-3.  Google Scholar

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