Multiple Homoclinic Solutions for Second‐Order Perturbed Hamiltonian Systems. Issue 2 (25th June 2013) - Ye, Yiwei; Tang, Chun‐Lei
<x xml:space="preserve">Abstract</x> In this paper, we study the second‐order perturbed Hamiltonian systems <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg42b8xn9p" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />ü(t)−λL(t)u(t)+∇W(t, u(t))=f(t), </alternatives> where <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg42b8xn5g" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />λ≥1</alternatives> is a parameter, <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg42b8xn61" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />L(t)</alternatives> is positive definite for all <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg42b8xn7k" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />t∈R</alternatives> but unnecessarily uniformly positive definite for <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg42b8xn84" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />t∈R</alternatives>, and W is either asymptotically quadratic or superquadratic in x as <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg42b8xnb7" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />|x|→∞</alternatives>. Based on variational methods, we prove the existence of at least two nontrivial homoclinic solutions for the above system when <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg42b8xncs" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />f∈L2(R, RN)∖0</alternatives> small enough.