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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 and 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), 639-642. doi: 10.1016/S0893-9659(03)00059-4.

[2]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381. doi: 10.1016/0022-1236(73)90051-7.

[3]

P. Caldiroli and P. Montecchiari, Homoclinic orbits for second order Hamiltonian systems with potential changing sign, Comm. Appl. Nonlinear Anal., 1 (1994), 97-129.

[4]

V. Coti Zelati and P. H. Rabinowitz, Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials, J. Amer. Math. Soc., 4 (1991), 693-727. doi: 10.1090/S0894-0347-1991-1119200-3.

[5]

A. Daouas, Homoclinic solutions for superquadratic Hamiltonian systems without periodicity assumption, Nonlinear Anal., 74 (2011), 3407-3418. doi: 10.1016/j.na.2011.03.001.

[6]

Y. Ding, Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems, Nonlinear Anal., 25 (1995), 1095-1113. doi: 10.1016/0362-546X(94)00229-B.

[7]

Y. Ding and C. Lee, Homoclinics for asymptotically quadratic and supequadratic Hamiltonian systems, Nonlinear Anal., 71 (2009), 1395-1413. doi: 10.1016/j.na.2008.10.116.

[8]

M. Izydorek and J. Janczewska, Homoclinic solutions for a class of the second order Hamiltonian systems, J. Differential Equations, 219 (2005), 375-389. doi: 10.1016/j.jde.2005.06.029.

[9]

P. Korman and A. C. Lazer, Homoclinic orbits for a class of symmetric Hamiltonian systems, Electron. J. Differential Equations, (1994), 1-10. doi: 10.1.1.27.6093.

[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), 1152-1158. doi: 10.1016/j.nonrwa.2011.09.008.

[11]

X. Lv, S. Lu and P. Yan, Existence of homoclinic solutions for a class of second-order Hamiltonian systems, Nonlinear Anal., 72 (2010), 390-398. doi: 10.1016/j.na.2009.06.073.

[12]

Y. Lv and C. Tang, Existence of even homoclinic orbits for a class of Hamiltonian systems, Nonlinear Anal., 67 (2007), 2189-2198. doi: 10.1016/j.na.2006.08.043.

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W. Omana and M. Willem, Homoclinic orbits for a class of Hamiltonian systems, Differential Integral Equations, 5 (1992), 1115-1120.

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H. Poincaré, Les méthodes nouvelles de la mécanique céleste, Gauthier-Villars, Pairs, 1897-1899.

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P. H. Rabinowitz, "Minimax Methods in Critical Point Theory with Applications to Differential Equations," CBMS Reg. Conf. Ser. in. Math., vol. 65, American Mathematical Society, Provodence, RI, 1986.

[16]

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.

[17]

P. H. Rabinowitz and K. Tanaka, Some results on connecting orbits for a class of Hamiltonian systems, Math. Z., 206 (1991), 473-499. doi: 10.1007/BF02571356.

[18]

A. Salvatore, Homoclinic orbits for a special class of nonautonomous Hamiltonian systems, Proceedings of the Second World Congress of Nonlinear Analysis, Part 8 (Athens, 1996), Nonlinear Anal., 30 (1997), 4849-4857. doi: SO362-546X(97)00142-9.

[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), 20-29. doi: 10.1016/j.jmaa.2010.06.038.

[20]

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

[21]

X. Tang and X. Lin, Infinitely many homoclinic orbits for Hamiltonian systems with indefinite subquadratic potentials, Nonlinear Anal., 74 (2011), 6314-6325. doi: 10.1016/j.na.2011.06.010.

[22]

X. Tang and X. Lin, Existence of infinitely many homoclinic orbits in Hamiltonian systems, Proc. Roy. Soc. Edinburgh Sect. A., 141 (2011), 1103-1119. doi: 10.1017/S0308210509001346.

[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), 255-271. doi: 10.3934/dcdsb.2011.15.255.

[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), 269-286. doi: 10.3934/cpaa.2011.10.269.

[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), 569-581. doi: 10.1016/j.jmaa.2010.01.060.

[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), 694-699. doi: 10.1016/j.jmaa.2009.12.024.

[27]

X. Wu and W. Zhang, Existence and multiplicity of homoclinic solutions for a class of damped vibration problems, Nonlinear Anal., 74 (2011), 4392-4398. doi: 10.1016/j.na.2011.03.059.

[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), 2742-2751. doi: 10.1016/j.nonrwa.2011.03.019.

[29]

M. Yang and Z. Han, Infinitely many homoclinic solutions for second-order Hamiltonian systems with odd nonlinearities, Nonlinear Anal., 74 (2011), 2635-2646. doi: 10.1016/j.na.2010.12.019.

[30]

R. Yuan and Z. Zhang, Homoclinic solutions for a class of second order non-autonomous systems, Electron. J. of Differential Equations, 128 (2009), 1-9.

[31]

Q. Zhang and C. Liu, Infinitely many homoclinic solutions for second order Hamiltonian systems, Nonlinear Anal., 72 (2010), 894-903. doi: 10.1016/j.na.2009.07.021.

[32]

Z. Zhang and R. Yuan, Homoclinic solutions for a class of non-autonomous subquadratic second order Hamiltonian systems, Nonlinear Anal., 71 (2009), 4125-4130. doi: 10.1016/j.na.2009.02.071.

[33]

Z. Zhang and R. Yuan, Homoclinic solutions of some second order non-autonomous systems, Nonlinear Anal., 71 (2009), 5790-5798. doi: 10.1016/j.na.2009.05.003.

[34]

Z. Zhang and R. Yuan, Homoclinic solutions for a class of asymptotically quadratic Hamiltonian systems, Nonlinear Analysis: Real World Applications, 11 (2010), 4185-4193. doi: 10.1016/j.nonrwa.2010.05.005.

[35]

W. Zhu, Existence of homoclinic solutions for a class of second order systems, Nonlinear Anal., 75 (2012), 2455-2463. doi: 10.1016/j.na.2011.10.043.

[36]

W. Zou and S. Li, Infinitely many homoclinic orbits for the second-order Hamiltonian systems, Appl. Math. Lett., 16 (2003), 1283-1287. doi: 10.1016/S0893-9659(03)90130-3.

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), 639-642. doi: 10.1016/S0893-9659(03)00059-4.

[2]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381. doi: 10.1016/0022-1236(73)90051-7.

[3]

P. Caldiroli and P. Montecchiari, Homoclinic orbits for second order Hamiltonian systems with potential changing sign, Comm. Appl. Nonlinear Anal., 1 (1994), 97-129.

[4]

V. Coti Zelati and P. H. Rabinowitz, Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials, J. Amer. Math. Soc., 4 (1991), 693-727. doi: 10.1090/S0894-0347-1991-1119200-3.

[5]

A. Daouas, Homoclinic solutions for superquadratic Hamiltonian systems without periodicity assumption, Nonlinear Anal., 74 (2011), 3407-3418. doi: 10.1016/j.na.2011.03.001.

[6]

Y. Ding, Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems, Nonlinear Anal., 25 (1995), 1095-1113. doi: 10.1016/0362-546X(94)00229-B.

[7]

Y. Ding and C. Lee, Homoclinics for asymptotically quadratic and supequadratic Hamiltonian systems, Nonlinear Anal., 71 (2009), 1395-1413. doi: 10.1016/j.na.2008.10.116.

[8]

M. Izydorek and J. Janczewska, Homoclinic solutions for a class of the second order Hamiltonian systems, J. Differential Equations, 219 (2005), 375-389. doi: 10.1016/j.jde.2005.06.029.

[9]

P. Korman and A. C. Lazer, Homoclinic orbits for a class of symmetric Hamiltonian systems, Electron. J. Differential Equations, (1994), 1-10. doi: 10.1.1.27.6093.

[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), 1152-1158. doi: 10.1016/j.nonrwa.2011.09.008.

[11]

X. Lv, S. Lu and P. Yan, Existence of homoclinic solutions for a class of second-order Hamiltonian systems, Nonlinear Anal., 72 (2010), 390-398. doi: 10.1016/j.na.2009.06.073.

[12]

Y. Lv and C. Tang, Existence of even homoclinic orbits for a class of Hamiltonian systems, Nonlinear Anal., 67 (2007), 2189-2198. doi: 10.1016/j.na.2006.08.043.

[13]

W. Omana and M. Willem, Homoclinic orbits for a class of Hamiltonian systems, Differential Integral Equations, 5 (1992), 1115-1120.

[14]

H. Poincaré, Les méthodes nouvelles de la mécanique céleste, Gauthier-Villars, Pairs, 1897-1899.

[15]

P. H. Rabinowitz, "Minimax Methods in Critical Point Theory with Applications to Differential Equations," CBMS Reg. Conf. Ser. in. Math., vol. 65, American Mathematical Society, Provodence, RI, 1986.

[16]

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.

[17]

P. H. Rabinowitz and K. Tanaka, Some results on connecting orbits for a class of Hamiltonian systems, Math. Z., 206 (1991), 473-499. doi: 10.1007/BF02571356.

[18]

A. Salvatore, Homoclinic orbits for a special class of nonautonomous Hamiltonian systems, Proceedings of the Second World Congress of Nonlinear Analysis, Part 8 (Athens, 1996), Nonlinear Anal., 30 (1997), 4849-4857. doi: SO362-546X(97)00142-9.

[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), 20-29. doi: 10.1016/j.jmaa.2010.06.038.

[20]

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

[21]

X. Tang and X. Lin, Infinitely many homoclinic orbits for Hamiltonian systems with indefinite subquadratic potentials, Nonlinear Anal., 74 (2011), 6314-6325. doi: 10.1016/j.na.2011.06.010.

[22]

X. Tang and X. Lin, Existence of infinitely many homoclinic orbits in Hamiltonian systems, Proc. Roy. Soc. Edinburgh Sect. A., 141 (2011), 1103-1119. doi: 10.1017/S0308210509001346.

[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), 255-271. doi: 10.3934/dcdsb.2011.15.255.

[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), 269-286. doi: 10.3934/cpaa.2011.10.269.

[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), 569-581. doi: 10.1016/j.jmaa.2010.01.060.

[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), 694-699. doi: 10.1016/j.jmaa.2009.12.024.

[27]

X. Wu and W. Zhang, Existence and multiplicity of homoclinic solutions for a class of damped vibration problems, Nonlinear Anal., 74 (2011), 4392-4398. doi: 10.1016/j.na.2011.03.059.

[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), 2742-2751. doi: 10.1016/j.nonrwa.2011.03.019.

[29]

M. Yang and Z. Han, Infinitely many homoclinic solutions for second-order Hamiltonian systems with odd nonlinearities, Nonlinear Anal., 74 (2011), 2635-2646. doi: 10.1016/j.na.2010.12.019.

[30]

R. Yuan and Z. Zhang, Homoclinic solutions for a class of second order non-autonomous systems, Electron. J. of Differential Equations, 128 (2009), 1-9.

[31]

Q. Zhang and C. Liu, Infinitely many homoclinic solutions for second order Hamiltonian systems, Nonlinear Anal., 72 (2010), 894-903. doi: 10.1016/j.na.2009.07.021.

[32]

Z. Zhang and R. Yuan, Homoclinic solutions for a class of non-autonomous subquadratic second order Hamiltonian systems, Nonlinear Anal., 71 (2009), 4125-4130. doi: 10.1016/j.na.2009.02.071.

[33]

Z. Zhang and R. Yuan, Homoclinic solutions of some second order non-autonomous systems, Nonlinear Anal., 71 (2009), 5790-5798. doi: 10.1016/j.na.2009.05.003.

[34]

Z. Zhang and R. Yuan, Homoclinic solutions for a class of asymptotically quadratic Hamiltonian systems, Nonlinear Analysis: Real World Applications, 11 (2010), 4185-4193. doi: 10.1016/j.nonrwa.2010.05.005.

[35]

W. Zhu, Existence of homoclinic solutions for a class of second order systems, Nonlinear Anal., 75 (2012), 2455-2463. doi: 10.1016/j.na.2011.10.043.

[36]

W. Zou and S. Li, Infinitely many homoclinic orbits for the second-order Hamiltonian systems, Appl. Math. Lett., 16 (2003), 1283-1287. doi: 10.1016/S0893-9659(03)90130-3.

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