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

Summary: We consider the existence of homoclinic solutions for the following second-order non-autonomous Hamiltonian system:

$\stackrel{¨}{q}-L\left(t\right)q+{W}_{q}\left(t,q\right)=0,\phantom{\rule{2.em}{0ex}}\left(\mathrm{HS}\right)$

where $L\left(t\right)\in C\left(ℝ,{ℝ}^{{n}^{2}}\right)$ is a symmetric and positive definite matrix for all $t\in ℝ$, $W\left(t,q\right)={a\left(t\right)|q|}^{\gamma }$ with $a\left(t\right):ℝ\to {ℝ}^{+}$ 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 of ODE 37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods 47J30 Variational methods (nonlinear operator equations)
##### Keywords:
homoclinic solutions; critical point; variational methods