January  2016, 15(1): 57-72. doi: 10.3934/cpaa.2016.15.57

Existence and nonuniqueness of homoclinic solutions for second-order Hamiltonian systems with mixed nonlinearities

1. 

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

Received  December 2014 Revised  October 2015 Published  December 2015

In this paper, we study the existence of homoclinic solutions to the following second-order Hamiltonian systems \begin{eqnarray} \ddot{u}(t)-L(t)u(t)+\nabla W(t,u(t))=0,\quad \forall t\in R, \end{eqnarray} where $L(t)$ is a symmetric and positive definite matrix for all $t\in R$. The nonlinear potential $W$ is a combination of superlinear and sublinear terms. By different conditions on the superlinear and sublinear terms, we obtain existence and nonuniqueness of nontrivial homoclinic solutions to above systems.
Citation: Dong-Lun Wu, Chun-Lei Tang, Xing-Ping Wu. Existence and nonuniqueness of homoclinic solutions for second-order Hamiltonian systems with mixed nonlinearities. Communications on Pure and Applied Analysis, 2016, 15 (1) : 57-72. doi: 10.3934/cpaa.2016.15.57
References:
[1]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.

[2]

A. Ambrosetti and V. Coti Zelati, Multiple homoclinic orbits for a class of conservative systems, Rend. Sem. Mat. Univ. Padova, 89 (1993), 177-194.

[3]

K. C. Chang, Infinite Dimensional Morse Theory and Multiple Solution Problems, Birkhäuser, Boston 1993. doi: 10.1007/978-1-4612-0385-8.

[4]

P. C. Carrião and O. H. Miyagaki, Existence of homoclinic solutions for a class of time-dependent Hamiltonian systems, J. Math. Anal. Appl., 230 (1999), 157-172. doi: 10.1006/jmaa.1998.6184.

[5]

H. W. Chen and Z. M. He, Infinitely many homoclinic solutions for a class of second-order Hamiltonian systems, Advances in Difference Equations, 2014 (2014):161.

[6]

Y. H. 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. H. Ding and S. J. Li, Homoclinic orbits for first order Hamiltonian systems, J. Math. Anal. Appl., 189 (1995), 585-601. doi: 10.1006/jmaa.1995.1037.

[8]

P. L. Felmer and Elves A. De B. Silva, Homoclinic and periodic orbits for Hamiltonian systems, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 26 (1998), 285-301.

[9]

G. H. Fei, The existence of homoclinic orbits for Hamiltonian systems with the potentials changing sign, Chinese Ann. Math. Ser. B, 17 (1996), 403-410.

[10]

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

[11]

S. P. Lu, Homoclinic solutions for a nonlinear second order differential system with p-Laplacian operator, Nonlinear Anal. RWA., 12 (2011), 525-534. doi: 10.1016/j.nonrwa.2010.06.037.

[12]

Y. Lv and C.-L. Tang, Homoclinic orbits for second-order Hamiltonian systems with subquadratic potentials, Chaos, Solitons & Fractals, 57 (2013), 137-145. doi: 10.1016/j.chaos.2013.09.007.

[13]

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

[14]

Z.-Q. Ou and C.-L. Tang, Existence of homoclinic solution for the second order Hamiltonian systems, J. Math. Anal. Appl., 291 (2004), 203-213. doi: 10.1016/j.jmaa.2003.10.026.

[15]

E. Paturel, Multiple homoclinic orbits for a class of Hamiltonian systems, Calc. Var. Partial Differential Equations, 12 (2001), 117-143. doi: 10.1007/PL00009909.

[16]

P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, in CBMS, Regional Conf. Ser. in Math., vol. 65, Amer. Math. Soc., Providence, RI, 1986.

[17]

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.

[18]

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.

[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]

X. H. Tang and X. Y. Lin, Existence of infinitely many homoclinic orbits in Hamiltonian systems, Pro. Roy. Soc. Edin., 141 (2011), 1103-1119. doi: 10.1017/S0308210509001346.

[21]

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

[22]

L.-L. Wan and C.-L. Tang, Existence of homoclinic orbits for second order Hamiltonian systems without (AR) condition, Nonlinear Anal., 74 (2011), 5303-5313. doi: 10.1016/j.na.2011.05.011.

[23]

L. Yang, H. Chen and J. Sun, Infinitely many homoclinic solutions for some second order Hamiltonian systems, Nonlinear Anal., 74 (2011), 6459-6468. doi: 10.1016/j.na.2011.06.029.

[24]

Y. W. Y and C.-L. Tang, Multiple homoclinic solutions for second-order perturbed Hamiltonian systems, Studies in Applied Mathematics, 132 (2014), 112-137. doi: 10.1111/sapm.12023.

[25]

R. Yuan and Z. Zhang, Homoclinic solutions for a class of second order Hamiltonian systems, Results in Math., 61 (2012) 195-208. doi: 10.1007/s00025-010-0088-3.

[26]

M.-H Yang and Z.-Q. 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.

[27]

Z. Zhang, X. Tian and R. Yuan, Homoclinic solutions for subquadratic Hamiltonian systems without coercive conditions, Taiwanese J. Math., 18 (2014), 1089-1105. doi: 10.11650/tjm.18.2014.3508.

show all references

References:
[1]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.

[2]

A. Ambrosetti and V. Coti Zelati, Multiple homoclinic orbits for a class of conservative systems, Rend. Sem. Mat. Univ. Padova, 89 (1993), 177-194.

[3]

K. C. Chang, Infinite Dimensional Morse Theory and Multiple Solution Problems, Birkhäuser, Boston 1993. doi: 10.1007/978-1-4612-0385-8.

[4]

P. C. Carrião and O. H. Miyagaki, Existence of homoclinic solutions for a class of time-dependent Hamiltonian systems, J. Math. Anal. Appl., 230 (1999), 157-172. doi: 10.1006/jmaa.1998.6184.

[5]

H. W. Chen and Z. M. He, Infinitely many homoclinic solutions for a class of second-order Hamiltonian systems, Advances in Difference Equations, 2014 (2014):161.

[6]

Y. H. 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. H. Ding and S. J. Li, Homoclinic orbits for first order Hamiltonian systems, J. Math. Anal. Appl., 189 (1995), 585-601. doi: 10.1006/jmaa.1995.1037.

[8]

P. L. Felmer and Elves A. De B. Silva, Homoclinic and periodic orbits for Hamiltonian systems, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 26 (1998), 285-301.

[9]

G. H. Fei, The existence of homoclinic orbits for Hamiltonian systems with the potentials changing sign, Chinese Ann. Math. Ser. B, 17 (1996), 403-410.

[10]

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

[11]

S. P. Lu, Homoclinic solutions for a nonlinear second order differential system with p-Laplacian operator, Nonlinear Anal. RWA., 12 (2011), 525-534. doi: 10.1016/j.nonrwa.2010.06.037.

[12]

Y. Lv and C.-L. Tang, Homoclinic orbits for second-order Hamiltonian systems with subquadratic potentials, Chaos, Solitons & Fractals, 57 (2013), 137-145. doi: 10.1016/j.chaos.2013.09.007.

[13]

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

[14]

Z.-Q. Ou and C.-L. Tang, Existence of homoclinic solution for the second order Hamiltonian systems, J. Math. Anal. Appl., 291 (2004), 203-213. doi: 10.1016/j.jmaa.2003.10.026.

[15]

E. Paturel, Multiple homoclinic orbits for a class of Hamiltonian systems, Calc. Var. Partial Differential Equations, 12 (2001), 117-143. doi: 10.1007/PL00009909.

[16]

P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, in CBMS, Regional Conf. Ser. in Math., vol. 65, Amer. Math. Soc., Providence, RI, 1986.

[17]

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.

[18]

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.

[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]

X. H. Tang and X. Y. Lin, Existence of infinitely many homoclinic orbits in Hamiltonian systems, Pro. Roy. Soc. Edin., 141 (2011), 1103-1119. doi: 10.1017/S0308210509001346.

[21]

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

[22]

L.-L. Wan and C.-L. Tang, Existence of homoclinic orbits for second order Hamiltonian systems without (AR) condition, Nonlinear Anal., 74 (2011), 5303-5313. doi: 10.1016/j.na.2011.05.011.

[23]

L. Yang, H. Chen and J. Sun, Infinitely many homoclinic solutions for some second order Hamiltonian systems, Nonlinear Anal., 74 (2011), 6459-6468. doi: 10.1016/j.na.2011.06.029.

[24]

Y. W. Y and C.-L. Tang, Multiple homoclinic solutions for second-order perturbed Hamiltonian systems, Studies in Applied Mathematics, 132 (2014), 112-137. doi: 10.1111/sapm.12023.

[25]

R. Yuan and Z. Zhang, Homoclinic solutions for a class of second order Hamiltonian systems, Results in Math., 61 (2012) 195-208. doi: 10.1007/s00025-010-0088-3.

[26]

M.-H Yang and Z.-Q. 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.

[27]

Z. Zhang, X. Tian and R. Yuan, Homoclinic solutions for subquadratic Hamiltonian systems without coercive conditions, Taiwanese J. Math., 18 (2014), 1089-1105. doi: 10.11650/tjm.18.2014.3508.

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