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Pointwise estimates of solutions to conservation laws with nonlocal dissipation-type terms
Homoclinic orbits for a class of asymptotically quadratic Hamiltonian systems
School of Mathematics and Statistics, Southwest University, Chongqing 400715, China |
$ \ddot{q}(t)-\lambda q(t)+\nabla W(t,q(t)) = 0 $ |
$ \lambda>0 $ |
$ \frac{|\nabla W(t,x)|}{|x|} $ |
$ |x|\rightarrow\infty $ |
$ |t|\rightarrow\infty $ |
References:
[1] |
C. O. Alves, P. C. Carriao and O. H. Miyagaki,
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] |
G. Arioli and A. Szulkin, Homoclinic solution for a class of systems of second order differential equtions, Tech. Rep. 5, Dept. of Math., Univ. Stockholm, Sweden, 1995.
doi: 10.12775/TMNA.1995.040. |
[3] |
P. Bartolo, V. Benci and D. Fortunato,
Abstract critical point theorems and applications to some nonlinear problems with ``strong" resonance at infinity, Nonlinear Anal., 7 (1983), 981-1012.
doi: 10.1016/0362-546X(83)90115-3. |
[4] |
F. A. Berezin and M. A. Shubin, The Schrodinger Equation, Kluwer, Dordrecht, 1991.
doi: 10.1007/978-94-011-3154-4. |
[5] |
P. C. Carriao 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. |
[6] |
G. W. Chen,
Superquadratic or asymptotically quadratic Hamiltonian systems: ground state homoclinic orbits, Ann. Mat. Pura Appl., 194 (2015), 903-918.
doi: 10.1007/s10231-014-0403-9. |
[7] |
D. G. Coata and C. A. Magalhães,
Variational elliptic problems which are nonquadratic at infinity, Nonlinear Anal., 23 (1994), 1401-1412.
doi: 10.1016/0362-546X(94)90135-X. |
[8] |
D. G. Costa and H. Tehrani,
On a class of asymptotically linear elliptic problems in $R^{N}$, J. Differential Equations, 173 (2001), 470-494.
doi: 10.1006/jdeq.2000.3944. |
[9] |
V. Coti-Zelati, I. Ekeland and E. Sere,
A variational approach to homoclinic orbits in Hamiltonian systems, Math. Ann., 288 (1990), 133-160.
doi: 10.1007/BF01444526. |
[10] |
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. |
[11] |
Y. Ding and C. Lee,
Homoclinics for asymptotically quadratic and superquadratic Hamiltonian systems, Nonlinear Anal., 71 (2009), 1395-1413.
doi: 10.1016/j.na.2008.10.116. |
[12] |
G. H. Fei,
The existence of homoclinic orbits for Hamiltonian systems with the potentials changing sign, Chinese Ann. Math. Ser. A, 17 (1996), 403-410.
|
[13] |
P. L. Felmer and E. A. De B. e. Silva,
Homoclinic and periodic orbits for Hamiltonian systems, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 26 (1998), 285-301.
|
[14] |
P. Korman and A. C. Lazer,
Homoclinic orbits for a class of symmetric Hamiltonian systems, Electronic J. Differential Equations, 1994 (1994), 1-10.
|
[15] |
P. L. Lions,
The concentration-compactness principle in the calculus of variations, The locally compact case. Ⅰ., Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145.
|
[16] |
P. L. Lions,
The concentration-compactness principle in the calculus of variations, The locally compact case. Ⅱ, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223-283.
|
[17] |
Z. Liu, S. Guo and Z. Zhang,
Homoclinic orbits for the second-order Hamiltonian systems, Nonlinear Anal. Real World Appl., 36 (2017), 116-138.
doi: 10.1016/j.nonrwa.2016.12.006. |
[18] |
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. |
[19] |
Y. Lv and C. L. Tang,
Existence of even homoclinic orbits for second-order Hamiltonian systems, Nonlinear Anal., 67 (2007), 2189-2198.
doi: 10.1016/j.na.2006.08.043. |
[20] |
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. |
[21] |
I. Marek and J. Joanna,
Homoclinic solutions for a class of second order Hamiltonian systems, J. Differential. Equations, 219 (2005), 375-389.
doi: 10.1016/j.jde.2005.06.029. |
[22] |
W. Omana and M. Willem,
Homoclinic orbits for a class of Hamiltonian systems, Differential Integral Equations, 5 (1992), 1115-1120.
|
[23] |
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. |
[24] |
P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, in CBMS Regional Conference Series in Mathematics, 65. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1986.
doi: 10.1090/cbms/065. |
[25] |
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. |
[26] |
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. |
[27] |
A. Salvatore,
Homoclinic orbits for a special class of nonautonomous Hamiltonian systems, Nonlinear Anal., 30 (1997), 4849-4857.
doi: 10.1016/S0362-546X(97)00142-9. |
[28] |
E. Serra, M. Tarallo and S. Terracini,
Subharmonic solutions to second-order differential equations with periodic nonlinearities, Nonlinear Anal., 41 (2000), 649-667.
doi: 10.1016/S0362-546X(98)00302-2. |
[29] |
J. Sun, H. Chen and Juan 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. |
[30] |
M. Yang and Z. Han,
Infinitly homoclinic solutions for second-order Hamiltonian systems with odd nonlinearities, Nonlinear Anal., 74 (2011), 2635-2646.
doi: 10.1016/j.na.2010.12.019. |
[31] |
Y. Ye and C. L. Tang,
Multiple homoclinic solutions for second-order perturbed Hamiltonian systems, Stud. Appl. Math., 132 (2014), 112-137.
doi: 10.1111/sapm.12023. |
[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] |
Q. Zhang and X. H. Tang,
Existence of homoclinic solutions for a class of asymptotically quadratic non-autonomous Hamiltonian systems, Math. Nachr., 285 (2012), 778-789.
doi: 10.1002/mana.201000096. |
[34] |
Q. Zheng,
Homoclinic solutions for a second-order nonperiodic asymptotically linear Hamiltonian systems, Abstr. Appl. Anal., 7 (2013), 34-37.
doi: 10.1155/2013/417020. |
[35] |
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. Carriao and O. H. Miyagaki,
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] |
G. Arioli and A. Szulkin, Homoclinic solution for a class of systems of second order differential equtions, Tech. Rep. 5, Dept. of Math., Univ. Stockholm, Sweden, 1995.
doi: 10.12775/TMNA.1995.040. |
[3] |
P. Bartolo, V. Benci and D. Fortunato,
Abstract critical point theorems and applications to some nonlinear problems with ``strong" resonance at infinity, Nonlinear Anal., 7 (1983), 981-1012.
doi: 10.1016/0362-546X(83)90115-3. |
[4] |
F. A. Berezin and M. A. Shubin, The Schrodinger Equation, Kluwer, Dordrecht, 1991.
doi: 10.1007/978-94-011-3154-4. |
[5] |
P. C. Carriao 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. |
[6] |
G. W. Chen,
Superquadratic or asymptotically quadratic Hamiltonian systems: ground state homoclinic orbits, Ann. Mat. Pura Appl., 194 (2015), 903-918.
doi: 10.1007/s10231-014-0403-9. |
[7] |
D. G. Coata and C. A. Magalhães,
Variational elliptic problems which are nonquadratic at infinity, Nonlinear Anal., 23 (1994), 1401-1412.
doi: 10.1016/0362-546X(94)90135-X. |
[8] |
D. G. Costa and H. Tehrani,
On a class of asymptotically linear elliptic problems in $R^{N}$, J. Differential Equations, 173 (2001), 470-494.
doi: 10.1006/jdeq.2000.3944. |
[9] |
V. Coti-Zelati, I. Ekeland and E. Sere,
A variational approach to homoclinic orbits in Hamiltonian systems, Math. Ann., 288 (1990), 133-160.
doi: 10.1007/BF01444526. |
[10] |
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. |
[11] |
Y. Ding and C. Lee,
Homoclinics for asymptotically quadratic and superquadratic Hamiltonian systems, Nonlinear Anal., 71 (2009), 1395-1413.
doi: 10.1016/j.na.2008.10.116. |
[12] |
G. H. Fei,
The existence of homoclinic orbits for Hamiltonian systems with the potentials changing sign, Chinese Ann. Math. Ser. A, 17 (1996), 403-410.
|
[13] |
P. L. Felmer and E. A. De B. e. Silva,
Homoclinic and periodic orbits for Hamiltonian systems, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 26 (1998), 285-301.
|
[14] |
P. Korman and A. C. Lazer,
Homoclinic orbits for a class of symmetric Hamiltonian systems, Electronic J. Differential Equations, 1994 (1994), 1-10.
|
[15] |
P. L. Lions,
The concentration-compactness principle in the calculus of variations, The locally compact case. Ⅰ., Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145.
|
[16] |
P. L. Lions,
The concentration-compactness principle in the calculus of variations, The locally compact case. Ⅱ, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223-283.
|
[17] |
Z. Liu, S. Guo and Z. Zhang,
Homoclinic orbits for the second-order Hamiltonian systems, Nonlinear Anal. Real World Appl., 36 (2017), 116-138.
doi: 10.1016/j.nonrwa.2016.12.006. |
[18] |
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. |
[19] |
Y. Lv and C. L. Tang,
Existence of even homoclinic orbits for second-order Hamiltonian systems, Nonlinear Anal., 67 (2007), 2189-2198.
doi: 10.1016/j.na.2006.08.043. |
[20] |
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. |
[21] |
I. Marek and J. Joanna,
Homoclinic solutions for a class of second order Hamiltonian systems, J. Differential. Equations, 219 (2005), 375-389.
doi: 10.1016/j.jde.2005.06.029. |
[22] |
W. Omana and M. Willem,
Homoclinic orbits for a class of Hamiltonian systems, Differential Integral Equations, 5 (1992), 1115-1120.
|
[23] |
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. |
[24] |
P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, in CBMS Regional Conference Series in Mathematics, 65. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1986.
doi: 10.1090/cbms/065. |
[25] |
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. |
[26] |
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. |
[27] |
A. Salvatore,
Homoclinic orbits for a special class of nonautonomous Hamiltonian systems, Nonlinear Anal., 30 (1997), 4849-4857.
doi: 10.1016/S0362-546X(97)00142-9. |
[28] |
E. Serra, M. Tarallo and S. Terracini,
Subharmonic solutions to second-order differential equations with periodic nonlinearities, Nonlinear Anal., 41 (2000), 649-667.
doi: 10.1016/S0362-546X(98)00302-2. |
[29] |
J. Sun, H. Chen and Juan 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. |
[30] |
M. Yang and Z. Han,
Infinitly homoclinic solutions for second-order Hamiltonian systems with odd nonlinearities, Nonlinear Anal., 74 (2011), 2635-2646.
doi: 10.1016/j.na.2010.12.019. |
[31] |
Y. Ye and C. L. Tang,
Multiple homoclinic solutions for second-order perturbed Hamiltonian systems, Stud. Appl. Math., 132 (2014), 112-137.
doi: 10.1111/sapm.12023. |
[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] |
Q. Zhang and X. H. Tang,
Existence of homoclinic solutions for a class of asymptotically quadratic non-autonomous Hamiltonian systems, Math. Nachr., 285 (2012), 778-789.
doi: 10.1002/mana.201000096. |
[34] |
Q. Zheng,
Homoclinic solutions for a second-order nonperiodic asymptotically linear Hamiltonian systems, Abstr. Appl. Anal., 7 (2013), 34-37.
doi: 10.1155/2013/417020. |
[35] |
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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