Homoclinic solutions for a class of non-autonomous subquadratic second-order Hamiltonian systems. (English) Zbl 1173.34330

Zhang, Ziheng; Yuan, Rong

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 71, No. 9, 4125-4130 (2009).

Summary: We consider the existence of homoclinic solutions for the following second-order non-autonomous Hamiltonian system:
\[ \ddot q-L(t)q+W_q(t,q)=0,\tag{HS} \]
where \(L(t)\in C(\mathbb R,\mathbb R^{n^2})\) is a symmetric and positive definite matrix for all \(t\in\mathbb R\), \(W(t,q)=a(t)|q|^\gamma\) with \(a(t):\mathbb R\to\mathbb R^+\) source is a positive continuous function and \(1<\gamma<2\) is a constant. Adopting some other reasonable assumptions for \(L\) and \(W\), we obtain a new criterion for guaranteeing that (HS) has one nontrivial homoclinic solution by use of a standard minimizing argument in critical point theory. Recent results from the literature are generalized and significantly improved.

MSC:

34C37	Homoclinic and heteroclinic solutions to ordinary differential equations
37J45	Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
47J30	Variational methods involving nonlinear operators

Keywords:

homoclinic solutions; critical point; variational methods

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