## Existence and multiplicity of homoclinic orbits for the second order Hamiltonian systems.(English)Zbl 1204.37067

The paper is concerned with the existence of homoclinic orbits of the second order Hamiltonian system $-\ddot{u}+L(t) u=\nabla_u R(t,u) \tag{HS}$ in $$\mathbb{R}^N$$. Here $$L(t)\in\mathbb{R}^{N\times N}$$ is symmetric and continuous in $$t$$, and $$R:\mathbb{R}\times\mathbb{R}^n\to\mathbb{R}$$ is $$C^1$$ and satisfies $$0\leq R(t,u)=o(|u|)$$ as $$u\to 0$$. It is allowed that $$\inf_{t\in\mathbb{R}}\min\,\mathrm{spec}L(t)=-\infty$$ so that the operator $$-\frac{d^2}{dt^2}+L(t)$$ may not be bounded below, and the associated quadratic form on $$L^2(\mathbb{R},\mathbb{R}^n)$$ is strongly indefinite. Assuming that $$R_u$$ is asymptotically linear in $$u$$, the authors present conditions on $$L$$ and $$R$$ such that (HS) has a homoclinic solution. If $$R$$ is even in $$u$$, they obtain multiple homoclinics. The proofs are based on a generalized linking theorem for strongly indefinite functionals due to Y. Ding and the reviewer [Math. Nachr. 279, No. 12, 1267–1288 (2006; Zbl 1117.58007)].

### MSC:

 37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)

Zbl 1117.58007
Full Text:

### References:

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