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)
Homoclinic orbits for asymptotically linear Hamiltonian systems. (English) Zbl 0984.37072
The existence of a homoclinic orbit is proved in the paper for a Hamiltonian system $$ \dot z=JH_z(z,t),\tag{1} $$ where $z=(p,q)\in \Bbb R^{2N}$ and $J=\left (\smallmatrix 0 & -I\\ I & 0\endsmallmatrix \right)$. Furthermore, $H(z,t)=\frac{1}{2}Az\cdot z+G(z,t)$ and $H(0,t)=0$ with $G_z(z,t)/|z|\to 0$ uniformly in $t$ as $z\to 0$, $G$ is 1-periodic in $t$ and asymptotically linear at infinity, and $JA$ is a hyperbolic matrix. $G$ has additional properties. A variational method is used to get an abstract theorem which is applied for showing a homoclinic orbit of (1). That theorem is also used to show a decaying solution of an asymptotically linear Schrödinger equation $-\triangle u+V(x)u=f(x,u)$ for $x\in \Bbb R^N$, $V\in C(\Bbb R^n,\Bbb R)$ and $f\in C(\Bbb R^N,\Bbb R)$.

MSC:
37J45Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods
WorldCat.org
Full Text: DOI
References:
[1] Ambrosetti, A.; Zelati, V. Coti: Multiplicité des orbites homoclines pour des systèmes conservatifs. CR acad. Sci. Paris 314, 601-604 (1992) · Zbl 0780.49008
[2] Carrião, P. C.; Miyagaki, O. H.: Existence of homoclinic solutions for a class of time-dependent Hamiltonian systems. J. math. Anal. appl. 230, 157-172 (1999) · Zbl 0919.34046
[3] Zelati, V. Coti; Ekeland, I.; Séré, E.: A variational approach to homoclinic orbits in Hamiltonian systems. Math. ann. 228, 133-160 (1990) · Zbl 0731.34050
[4] Zelati, V. Coti; Rabinowitz, P. H.: Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials. J. amer. Math. soc. 4, 693-727 (1991) · Zbl 0744.34045
[5] Ding, Y.: Existence and multiciplicity results for homoclinic solutions to a class of Hamiltonian systems. Nonlinear anal. 25, 1095-1113 (1995) · Zbl 0840.34044
[6] Ding, Y.; Girardi, M.: Periodic and homoclinic solutions to a class of Hamiltonian systems with the potentials changing sign. Dynam. system appl. 2, 131-145 (1993) · Zbl 0771.34031
[7] Ding, Y.; Li, S.: Homoclinic orbits for first order Hamiltonian systems. J. math. Anal. appl. 189, 585-601 (1995) · Zbl 0818.34023
[8] Ding, Y.; Willem, M.: Homoclinic orbits of a Hamiltonian system. Z. angew. Math. phys. 50, 759-778 (1999) · Zbl 0997.37041
[9] Hofer, H.; Wysocki, K.: First order elliptic systems and the existence of homoclinic orbits in Hamiltonian systems. Math. ann. 228, 483-503 (1990) · Zbl 0702.34039
[10] Jeanjean, L.: On the existence of bounded palais--Smale sequences and application to a landesman--lazer type problem set on RN. Proc. roy. Soc. Edinburgh 129A, 787-809 (1999) · Zbl 0935.35044
[11] Kryszewski, W.; Szulkin, A.: Generalized linking theorem with an application to semilinear Schrödinger equation. Adv. differential equations 3, 441-472 (1998) · Zbl 0947.35061
[12] Lions, Pl: The concentration-compactness principle in the calculus of variations. The locally compact case. Part II. Ann. inst. H. Poincaré, anal. Non linéaire 1, 223-283 (1984) · Zbl 0704.49004
[13] Omana, W.; Willem, M.: Homoclinic orbits for a class of Hamiltonian systems. Differential integral equations 5, 1115-1120 (1992) · Zbl 0759.58018
[14] Rabinowitz, P. H.: Homoclinic orbits for a class of Hamiltonian systems. Proc. roy. Soc. Edinburgh 114, 33-38 (1990) · Zbl 0705.34054
[15] Rabinowitz, P. H.; Tanaka, K.: Some results on connecting orbits for a class of Hamiltonian systems. Math. Z. 206, 472-499 (1991) · Zbl 0707.58022
[16] Reed, M.; Simon, B.: Methods of modern mathematical physics. (1978) · Zbl 0401.47001
[17] Séré, E.: Existence of infinitely many homoclinic orbits in Hamiltonian systems. Math. Z. 209, 27-42 (1992) · Zbl 0725.58017
[18] Séré, E.: Looking for the Bernoulli shift. Ann. inst. H. Poincaré, anal. Non linéaire 10, 561-590 (1993)
[19] Stuart, C. A.: Bifurcation into spectral gaps. Bull. belg. Math. soc., 59 (1995) · Zbl 0864.47037
[20] 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
[21] Willem, M.: Minimax theorems. (1996) · Zbl 0856.49001