Existence and multiplicity of homoclinic orbits for a class of asymptotically periodic second order Hamiltonian systems
type: Item
instance of: scholarly article
title: Existence and multiplicity of homoclinic orbits for a class of asymptotically periodic second order Hamiltonian systems
zbMATH Open document ID: 0849.34035
DOI: 10.1007/BF01759265
author: Piero Montecchiari
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 1996-07-08T00:00:00Z
review text: The paper deals with the system \(d^2 q/dt^2 = L(t) q - \nabla V(t,q)\), \(q(t) \to 0\), \(dq (t)/dt \to 0\) as \(|t |\to \infty\), where \(q \in \mathbb{C}^2 (\mathbb{R}, \mathbb{R}^m)/ \{0\}\) and \(L \in \mathbb{C} (\mathbb{R}, \mathbb{R}^{m^2})\), \(V \in \mathbb{C}^1 (\mathbb{R}, \mathbb{R}^m)\) are asymptotic to certain periodic functions \(L_-\), \(V_-\) as \(t \to - \infty\). (More in detail: \(L(t)\), \(L_- (t)\) are symmetrical; \(ax^2 \geq L(t)x \cdot x\) and \(L_- (t)x \cdot x \geq bx^2\) for appropriate \(a,b > 0\); \(L_- (t + 1) = L_- (t)\); \(|L(t) - L_- (t) |\to 0\) as \(t \to - \infty\); \(V\) and \(V_-\) are \(o(x)\) as \(x \to 0\); \(0 < \mu V < \nabla V \cdot x\) and \(0 < \mu V_- < \nabla V_- \cdot x\) for appropriate \(\mu > 0\) and all \(x \neq 0\); \(V_- (t + 1,x) = V_- (t,x)\); \(|\nabla V(t,x) - \nabla V_- (t,x) |\to 0\) as \(t \to - \infty\) uniformly on compacts.) The solutions are found as the critical points of the functional \(\varphi (u) = {1 \over 2} |u |^2 - \int V(t,u) dt\) on the space \(W^{1,2} (\mathbb{R}, \mathbb{R}^m)\) with the norm \(|u |^2 = \int (|du/dt |^2 + L(t)u \cdot u) dt\) under the following additional hypothesis: there exists \(c^* > c\) such that the set \(K(c^*)/ \mathbb{Z}\) is finite, where \(c\) is the mountain pass level, and \(K(c^*)\) is the set of critical values which are less or equal to \(c^*\) of the functional \(\varphi_- (u) = {1 \over 2} \int (|du/dt |^2 + L_- (t)u \cdot u - V(t,u)) dt\). A rather thorough analysis results in the existence of even infinitely many homoclinic solutions \(u \neq 0\) with the following \((k\)-bump) property: for every \(r > 0\) and each finite sequence \(p_1, \dots, p_k \in \mathbb{Z}\) that verifies \(p_j < n_0 (r)\), \(p_j - p_{j + 1} > M(r) > 0\) (with appropriate \(n_0 (r)\) and \(M(r) > 0)\) there exists a solution \(\overline q\) such that \(\sup |\overline q(t) - q(t) |< r({1 \over 2} (p_j + p_{j + 1}) < t < {1 \over 2} (p_{j - 1} + p_j))\) where \(p_0 = \infty\)…
Mathematics Subject Classification ID: 34C37
Mathematics Subject Classification ID: 37J99
zbMATH DE Number: 878453
zbMATH Keywords: \(k\)-bump solution
zbMATH Keywords: Hamiltonian system
zbMATH Keywords: mountain pass level
zbMATH Keywords: infinitely many homoclinic solutions
reviewed by: Jan Chrastina
MaRDI profile type: Publication
cites work: Q4695262
cites work: Critical points and nonlinear variational problems
cites work: Q4288396
cites work: Closed orbits of fixed energy for a class of \(N\)-body problems
cites work: Homoclinic orbits on compact manifolds
cites work: The existence of homoclinic motions
cites work: Existence and multiplicity of homoclinic orbits for potentials on unbounded domains
cites work: A variational approach to homoclinic orbits in Hamiltonian systems
cites work: Homoclinic Orbits for Second Order Hamiltonian Systems Possessing Superquadratic Potentials
cites work: Homoclinic type solutions for a semilinear elliptic PDE on ℝn
cites work: First order elliptic systems and the existence of homoclinic orbits in Hamiltonian systems
cites work: The concentration-compactness principle in the calculus of variations. The locally compact case. I
cites work: The concentration-compactness principle in the calculus of variations. The locally compact case. II
cites work: Homoclinic orbits for a class of Hamiltonian systems
cites work: Some results on connecting orbits for a class of Hamiltonian systems
cites work: Existence of infinitely many homoclinic orbits in Hamiltonian systems
cites work: Looking for the Bernoulli shift
cites work: Homoclinic orbits in a first order superquadratic Hamiltonian system: Convergence of subharmonic orbits
description: scientific article; zbMATH DE number 878453