zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Homoclinics for asymptotically quadratic and superquadratic Hamiltonian systems. (English) Zbl 1168.58302
Summary: We find new conditions, which are different from those used in previous related studies, to ensure the existence of infinitely many homoclinic orbits for the second order Hamiltonian systems of the form $$-\ddot q = V_q (t,q).$$ Here, we assume that $V(t,q)$ depends periodically on $t$, and assume, on $q$, that $V(t,q)$ is asymptotically quadratic at $q=0$ and is, as $|q|\rightarrow \infty $, either asymptotically quadratic or superquadratic, as well as the new conditions.

MSC:
58E05Abstract critical point theory
58E50Applications of variational methods in infinite-dimensional spaces
WorldCat.org
Full Text: DOI
References:
[1] Ambrosetti, A.; Rabinowitz, P.: Dual variational methods in critical point theory and applications. J. funct. Anal. 14, 349-381 (1973) · Zbl 0273.49063
[2] Ambrosetti, A.; Zelati, V. Coti: Multiplicité des orbites homoclines pour des systémes conservatifs. C. R. Acad. sci. Paris sér. I math. 314, 601-604 (1992) · Zbl 0780.49008
[3] Arioli, G.; Szulkin, A.: Homoclinic solutions of Hamiltonian systems with symmetry. J. differential equations 158, 291-313 (1999) · Zbl 0944.37030
[4] Cerami, G.: Un criterio di esistenza per i punti critici su varietá illimitate. Istit. lombardo accad. Sci. lett. Rend. A 112, 332-336 (1978)
[5] Coti-Zelati, V.; Ekeland, I.; Séré, E.: A variational approach to homoclinic orbits in Hamiltonian systems. Math. ann. 288, 133-160 (1990) · Zbl 0731.34050
[6] Coti-Zelati, V.; Rabinowitz, P.: Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials. J. amer. Math. soc. 4, 693-727 (1991) · Zbl 0744.34045
[7] Ding, Y. H.: Multiple homoclinics in a Hamiltonian system with asymptotically or super linear terms. Commun. contemp. Math. 8, 453-480 (2006) · Zbl 1104.70013
[8] Ding, Y. H.; Luan, S. X.: Multiple solutions for a class of nonlinear Schrödinger equations. J. differential equations 207, 423-457 (2004) · Zbl 1072.35166
[9] Ding, Y. H.; Lee, Cheng: Multiple solutions of Schrödinger equations with indefinite linear part and super or asymptotically linear terms. J. differential equations 222, 137-163 (2006) · Zbl 1090.35077
[10] Ding, Y. H.; Willem, M.: Homoclinic orbits of a Hamiltonian system. Z. angew. Math. phys. 50, 759-778 (1999) · Zbl 0997.37041
[11] Ding, Y. H.; Girardi, M.: Infinitely many homoclinic orbits of a hamitonian system with symmetry. Nonlinear anal. TMA 38, 391-415 (1999) · Zbl 0938.37034
[12] Hofer, H.; Wysocki, K.: First order elliptic systems and the existence of homoclinic orbits in Hamiltonian systems. Math. ann. 288, 483-503 (1990) · Zbl 0702.34039
[13] Jeanjean, L.; Tanaka, K.: Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities. Calc. var. Partial differential equations 21, No. 3, 287-318 (2004) · Zbl 1060.35012
[14] Lions, P. L.: The concentration-compactness principle in the calculus of variations. The locally compact cases, part II. AIP anal. Non linéaire 1, 223-283 (1984) · Zbl 0704.49004
[15] Melnikov, V. K.: On the stability of the center for periodic perturbations. Trans. Moscow math. Soc. 12, 1-57 (1963)
[16] Poincaé, H.: LES méthodes nouvelles de la mécanique céleste. (1897--1899)
[17] Rabinowitz, P. H.: Homoclinic orbits for a class of Hamiltonian systems. Proc. roy. Soc. edingburgh sect. A 114, 33-38 (1990) · Zbl 0705.34054
[18] Rabinowitz, P. H.; Tanaka, K.: Some results on connecting orbits for a class of Hamiltonian systems. Math. Z. 206, 473-499 (1991) · Zbl 0707.58022
[19] Séré, E.: Existence of infinitely many homoclinic orbits in Hamiltonian systems. Math. Z. 209, 27-42 (1992) · Zbl 0725.58017
[20] Séré, E.: Looking for the Bernoulli shift. Ann. inst. H. poincaé anal. Non linéaire 10, 561-590 (1993)
[21] Szulkin, A.; Zou, W.: Homoclinic orbits for asymptotically linear Hamiltonian systems. J. funct. Anal. 187, 25-41 (2001) · Zbl 0984.37072
[22] Tanaka, K.: Homoclinic orbits in a first order superquadratic Hamiltonian system. Convergence of subharmonic orbits. J. differential equations 94, 315-339 (1991) · Zbl 0787.34041