Homoclinic solutions for second order Hamiltonian systems with general potentials

Year, pages: 2016, 887 - 900

In this paper we are concerned with the existence of infinitely many homoclinic solutions for the following second order non-autonomous Hamiltonian systems

$$ \ddot u(t)-L(t)u(t)+\nabla W(t,u(t))=0, \eqno(HS) $$

where $t\in \mathbb{R}$, $L\in C(\mathbb{R},\mathbb{R}ⁿ²)$ is a symmetric and positive definite matrix for all $t\in \mathbb{R}$, $W\in C¹(\mathbb{R}× \mathbb{R}ⁿ,\mathbb{R})$ and $\nabla W(t,u)$ is the gradient of $W$ at $u$. The novelty of this paper is that, assuming that $L$ meets some coercive condition and the potential $W$ is of the form $W(t,u)=W₁(t,u)+W₂(t,u)$, for the first time we show that (HS) possesses two different sequences of infinitely many homoclinic solutions via the Fountain theorem and the dual Fountain theorem such that the corresponding energy functional of (HS) goes to infinity and zero, respectively. Some recent results in the literature are generalized and significantly improved.

Zhang, Z., You, H., Yuan, R. 2016. Homoclinic solutions for second order Hamiltonian systems with general potentials. In Mathematica Slovaca, vol. 66, no.4, pp. 887-900. 0139-9918. DOI: https://doi.org/10.1515/ms-2015-0190

Zhang, Z., You, H., Yuan, R. (2016). Homoclinic solutions for second order Hamiltonian systems with general potentials. Mathematica Slovaca, 66(4), 887-900. 0139-9918. DOI: https://doi.org/10.1515/ms-2015-0190