Existence and multiplicity of homoclinic orbits for potentials on unbounded domains. (English) Zbl 0807.34058

Author’s abstract: “We study the system \(\ddot q+ V'(q)= 0\), where \(V\) is a potential with a strict local maximum at 0 and possibly with a singularity. First, using a minimizing argument, we can prove the existence of a homoclinic orbit when the component \(\Omega\) of \(\{x\in \mathbb{R}^ N: V(x)< V(0)\}\) containing 0 is an arbitrary open set; in the case \(\Omega\) unbounded we allow \(V(x)\) to go to 0 at infinity, although at a slow enough rate. Then we show that the presence of a singularity in \(\Omega\) implies that a homoclinic orbit can also be found via a minimax procedure and, comparing the critical levels of the functional associated to the system, we see that these two solutions are distinct whenever the singularity is ‘not too far’ from 0”.


34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
70K05 Phase plane analysis, limit cycles for nonlinear problems in mechanics
Full Text: DOI


[1] DOI: 10.1016/0022-247X(91)90107-B · Zbl 0737.58052
[2] DOI: 10.1007/BF01444526 · Zbl 0731.34050
[3] Ambrosetti, Rend. Sent. Univ. Padova 89 pp 177– (1993)
[4] Ambrosetti, Partial Differential Equations and Related Subjects (1992)
[5] Tanaka, Ann. Inst. H. Poincaré Anal. Non Linéaire 7 pp 427– (1990) · Zbl 0712.58026
[6] DOI: 10.1016/0022-0396(91)90095-Q · Zbl 0787.34041
[7] Séré, Ann. Inst. H. Poincaré Anal. Non Lineaire 10 pp 561– (1993) · Zbl 0803.58013
[8] DOI: 10.1007/BF02570817 · Zbl 0725.58017
[9] DOI: 10.1007/BF02571356 · Zbl 0707.58022
[10] Rabinowitz, Proc. Roy. Soc. Edinburgh Sect. A 114 pp 33– (1990) · Zbl 0705.34054
[11] Rabinowitz, Ann. Inst. H. Poincaré Anal. Non Linéaire 6 pp 331– (1989) · Zbl 0701.58023
[12] Poincaré, Les Methodes Nouvelles de la Méchanique Céleste (1897)
[13] Lions, Rev. Mat. Iberoamericana 1 pp 145– (1985) · Zbl 0704.49005
[14] DOI: 10.1090/S0002-9947-1975-0377983-1
[15] DOI: 10.1007/BF01444543 · Zbl 0702.34039
[16] DOI: 10.1090/S0894-0347-1991-1119200-3
[17] DOI: 10.1016/0022-1236(89)90078-5 · Zbl 0681.70018
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.