Abstract
In this paper, we study the second-order Hamiltonian systems u¨−L(t)u+∇W(t,u)=0,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\ddot{u}-L(t)u+\\nabla W(t,u)=0, $$\\end{document} where tin mathbb{R}, uin mathbb{R}^{N}, L and W depend periodically on t, 0 lies in a spectral gap of the operator -d^{2}/dt^{2}+L(t) and W(t,x) is locally superquadratic. Replacing the common superquadratic condition that lim_{|x|rightarrow infty }frac{W(t,x)}{|x|^{2}}=+infty uniformly in tin mathbb{R} by the local condition that lim_{|x|rightarrow infty }frac{W(t,x)}{|x|^{2}}=+infty a.e. tin J for some open interval Jsubset mathbb{R}, we prove the existence of one nontrivial homoclinic soluiton for the above problem.
Highlights
Introduction and main resultsConsider the second-order Hamiltonian systems u – L(t)u + ∇W (t, u) = 0, (1.1)where t ∈ R, u ∈ RN, L ∈ C(R, RN×N ) and W ∈ C1(R × RN, R) satisfies the following basic conditions:(W1) W is T-periodic in t and there exist constants C0 > 0 and p > 2 such that∇W (t, x) ≤ C0 1 + |x|p–1, ∀(t, x) ∈ R × RN .(W2) ∇W (t, x) = o(|x|) as x → 0 uniformly in t and W (t, x) ≥ 0 for all (t, x)
In this paper, we study the second-order Hamiltonian systems u – L(t)u + ∇W(t, u) = 0, where t ∈ R, u ∈ RN, L and W depend periodically on t, 0 lies in a spectral gap of the operator –d2/dt2 + L(t) and W(t, x) is locally superquadratic
R, we prove the existence of one nontrivial homoclinic soluiton for the above problem
Summary
Apply a variant generalized weak linking theorem for strongly indefinite functionals developed by Schechter and Zou (see [16]). This approach is not very satisfactory, since working with a family of perturbed functionals makes things unnecessary complicated. The key point in our proof is that, φ may has unbounded (PS) sequences, we can prove that all Cerami sequences of φ are bounded (see Lemma 2.4 below), and Theorem 1.1 follows directly from the generalized linking theorem (see [11]).
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