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)
Infinitely many homoclinic solutions for second-order Hamiltonian systems with odd nonlinearities. (English) Zbl 1218.37082
The authors prove the existence of infinitely many homoclinic (to 0) solutions for the second order system $\ddot u-L(t)u+W_u(t,u) = 0$ under the conditions that $L=L(t)$ is a symmetric matrix such that the smallest eigenvalue of $L(t)$ tends to infinity as $|t|\to\infty$, and the function $W$ is even in $u$ and either sub- or superquadratic as $u\to 0$ and $|u|\to\infty$. In the subquadratic case, an Ahmad-Lazer-Paul type condition is imposed and for superquadratic $W$ a condition which is weaker than that of Ambrosetti and Rabinowitz is assumed. The proofs are carried out by verifying the assumptions of two versions of the fountain theorem, respectively due to Bartsch and Willem, and to Zou. Necessary compactness of the Euler-Lagrange functional corresponding to the problem is a consequence of the fact that the smallest eigenvalue of $L(t)$ goes to infinity with $|t|$.

MSC:
 37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods 34C37 Homoclinic and heteroclinic solutions of ODE 58E05 Abstract critical point theory
Full Text:
References:
 [1] Rabinowitz, P. H.: Homoclinic orbits for a class of Hamiltonian systems, Proc. roy. Soc. Edinburgh 114, 33-38 (1990) · Zbl 0705.34054 · doi:10.1017/S0308210500024240 [2] 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 · doi:10.1007/BF02571356 [3] Omana, W.; Willem, M.: Homoclinic orbits for a class of Hamiltonian systems, Differential integral equations 5, 1115-1120 (1992) · Zbl 0759.58018 [4] Ding, Y.: Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems, Nonlinear anal. 25, 1095-1113 (1995) · Zbl 0840.34044 · doi:10.1016/0362-546X(94)00229-B [5] Ou, Z.; Tang, C.: Existence of homoclinic solution for the second order Hamiltonian systems, J. math. Anal. appl. 291, 203-213 (2004) · Zbl 1057.34038 · doi:10.1016/j.jmaa.2003.10.026 [6] Yang, J.; Zhang, F.: Infinitely many homoclinic orbits for the second order Hamiltonian systems with super-quadratic potentials, Nonlinear anal. RWA 10, 1417-1423 (2009) · Zbl 1162.34328 · doi:10.1016/j.nonrwa.2008.01.013 [7] Zhang, Q.; Liu, C.: Infinitely many homoclinic solutions for second order Hamiltonian systems, Nonlinear anal. 72, 894-903 (2010) · Zbl 1178.37063 · doi:10.1016/j.na.2009.07.021 [8] Bartsch, T.; Willem, M.: On an elliptic equation with concave and convex nonlinearities, Proc. amer. Math. soc. 123, 3555-3561 (1995) · Zbl 0848.35039 · doi:10.2307/2161107 [9] Han, Z.: 2${\pi}$-periodic solutions to N-dimensional systems of Duffing’s type (II), J. Qingdao univ. 7, 19-26 (1994) [10] Xavier, J.; Miyagaki, O.: Remarks on a resonant problem with unbounded nonlinearity, J. math. Anal. appl. 209, 255-273 (1997) · Zbl 0878.35046 · doi:10.1006/jmaa.1997.5374 [11] A. Szulkin, T. Weth, The method of Nehari manifold, 2010. http://www2.math.su.se/ andrzejs/Recent_publications/. · Zbl 1218.58010 [12] Zou, W.: Variant Fountain theorems and their applications, Manuscripta math. 104, 343-358 (2001) · Zbl 0976.35026 · doi:10.1007/s002290170032 [13] Willem, M.: Minimax theorems, (1996) · Zbl 0856.49001 [14] Zou, W.; Schechter, M.: Critical point theory and its applications, (2006) · Zbl 1125.58004 [15] Mawhin, J.; Willem, M.: Critical point theory and Hamiltonian systems, (1989) · Zbl 0676.58017 [16] Rabinowitz, P. H.: Minimax methods in critical point theory with applications to differential equations, (1986) · Zbl 0609.58002 [17] Zelati, V. Coti; Rabinowitz, P. H.: Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potential, J. amer. Math. soc. 4, 693-727 (1991) · Zbl 0744.34045 · doi:10.2307/2939286