## Solutions of nonperiodic super quadratic Hamiltonian systems.(English)Zbl 1346.49016

Summary: This paper concerns solutions for the Hamiltonian system: $$\dot{u}=\mathcal JH_u(t,u)$$, where $$H(t,u)=1/2Lu\cdot u+W(t,u), L$$ is a $$2N\times 2N$$ symmetric matrix, and $$W\in C^1(\mathbb R\times\mathbb R^{2N},\mathbb R)$$. We consider the case that $$0\notin \sigma _{c}( - (Jd/dt+L))$$ and $$W$$ satisfies some new generalized super quadratic condition different from the type of Ambrosetti-Rabinowitz. The method is variational: by virtue of some auxiliary system related to the “limit equation” of the Hamiltonian system, we first establish that the $$(C)_{c}$$-condition holds true for all $$c$$ less than the least energy of the limit equation. Then, using some recently developed weak linking theorem, we obtain a least energy solution of the Hamiltonian system.

### MSC:

 49J45 Methods involving semicontinuity and convergence; relaxation 70H05 Hamilton’s equations 70K44 Homoclinic and heteroclinic trajectories for nonlinear problems in mechanics 70G75 Variational methods for problems in mechanics
Full Text:

### References:

 [1] Mawhin, Critical Point Theory and Hamiltonian Systems (1989) [2] Coti-Zelati, A variational approach to homoclinic orbits in Hamiltonian systems, Mathematische Annalen 228 pp 133– (1990) · Zbl 0731.34050 [3] Hofer, First order elliptic systems and the existence of homoclinic orbits in Hamiltonian systems, Mathematische Annalen 228 pp 483– (1990) · Zbl 0702.34039 [4] Séré, Existence of infinitely many homoclinic orbits in Hamiltonian systems, Mathematische Zeitschrift 209 pp 27– (1992) · Zbl 0725.58017 [5] Tanaka, Homoclinic orbits in a first order superquadratic Hamiltonian system: convergence of subharmonic orbits, Journal of Differential Equations 94 pp 315– (1991) · Zbl 0787.34041 [6] Arioli, Homoclinic solutions of Hamiltonian systems with symmetry, Journal of Differential Equations 158 pp 291– (1999) · Zbl 0944.37030 [7] Séré, Looking for the Bernoulli shift, Annales de l’Institut Henri Poincaré Analyse Non Linéaire 10 pp 561– (1993) · Zbl 0803.58013 [8] Szulkin, Homoclinic orbits for asymptotically linear Hamiltonian systems, Journal of Functional Analysis 187 pp 25– (2001) · Zbl 0984.37072 [9] Ding, Multiple homoclinics in a Hamiltonian system with asymptotically or super linear terms, Communications in Contemporary Mathematics 8 pp 453– (2006) · Zbl 1104.70013 [10] Ding, Homoclinic orbits of a Hamiltonian system, Zeitschrift für angewandte Mathematik und Physik 50 pp 759– (1999) · Zbl 0997.37041 [11] Ding, Infinitely many homoclinic orbits of a Hamiltonian system with symmetry, Nonlinear Analysis 38 pp 391– (1999) · Zbl 0938.37034 [12] Ding, Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems, Nonlinear Analysis 25 (11) pp 1095– (1995) · Zbl 0840.34044 [13] Ding, Homoclinic orbits for first order Hamiltonian systems, Journal of Mathematical Analysis and Applications 189 pp 585– (1995) · Zbl 0818.34023 [14] Ding, Homoclinic orbits for a nonperiodic Hamiltonian system, Journal of Differential Equations 237 pp 473– (2007) · Zbl 1117.37032 [15] Bartsch, Deformation theorems on non-metrizable vector spaces and applications to critical point theory, Mathematische Nachrichten 279 pp 1267– (2006) · Zbl 1117.58007 [16] Ding, Existence and exponential decay of homoclinics in a nonperiodic superquadratic Hamiltonian system, Journal of Differential Equations 246 pp 2829– (2009) · Zbl 1162.70014 [17] Willem, On a Schrodinger equation with periodic potential and spectrum point zero, Indiana University Mathematics Journal 52 pp 109– (2003) · Zbl 1030.35068 [18] Ding, Variational Methods for Strongly Indefinite Problems, Interdisciplinary Math. Sci. 7 (2007) · Zbl 1133.49001 [19] Cerami, Un criterio di esistenza per punti critici su varieta illimitate, Rendiconti, Accademia di Scienze e Lettere, Istituto Lombardo 112 pp 332– (1978) [20] Ackermann, A nonlinear superposition principle and multibump solutions of periodic Schrödinger equations, Journal of Functional Analysis 234 pp 423– (2006) · Zbl 1126.35057
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.