## 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:
$\ddot q-L(t)q+W_q(t,q)=0,\tag{HS}$
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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### References:

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