We study the existence of homoclinic solutions for a class of generalized Lagrangian systems in the plane, with a C1-smooth potential with a single well of infinite depth at a point ξ and a unique strict global maximum 0 at the origin.Under a strong force condition around the singular point ξ, via minimization of an action integral, we will prove the existence of at least two geometrically distinct homoclinic solutions.

We will be concerned with the existence of homoclinics for second order Hamiltonian systems in R^N (N>2) given by Hamiltonians of the form H(t,q,p)=Φ(p)+V(t,q), where Φ is a G-function in the sense of Trudinger, V is C^2-smooth, periodic in the time variable, has a single well of infinite depth at a point ξ and a unique strict global maximum 0 at the origin. Under a strong force type condition aroud the singular point ξ, we prove...

We are concerned with a planar autonomous Hamiltonian system with a potential possessing a single well of infinite depth at a point X and a unique strict global maximum 0 at a point A. Under a strong force condition around the singularity X, via minimization of an action integral and using a shadowing chain lemma together with simple geometrical arguments, we prove the existence of infinitely many geometrically distinct homoclinic...

W niniejszym artykule rozważamy autonomiczny układ Hamiltonowski na płaszczyźnie z potencjałem, który ma punkt osobliwy (studnię nieskończonej głębokości) i maksimum globalne właściwe równe zero przyjmowane w dwóch różnych punktach płaszczyzny. Przy założeniu, że w otoczeniu punktu osobliwego potencjał spełnia warunek Gordona(gradient tego potencjału w otoczeniu punktu osobliwego jest tzw. silną siłą, ang. a strong force) dowodzimy...

We consider a conservative second order Hamiltonian system \ddot{q}+ ∇V(q)=0 in R3 with a potential V having a global maximum at the origin and a line l ∩ {0} = ∅ as a set of singular points. Under a certain compactness condition on V at infinity and a strong force condition at singular points we study, by the use of variational methods and geometrical arguments, the existence of homoclinic solutions of the system.