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In this paper, we consider of the following second-order Hamiltonian system \begin{equation*} \ddot{u}(t)-L(t)u(t)+\nabla W(t,u(t))=0,\qquad \forall t \in \mathbb{R}, \end{equation*} where $W(t,x)$ is subquadratic at infinity. With a competition condition, we establish the existence of homoclinic solutions by using the variational methods. In our theorem, the smallest eigenvalue function $l(t)$ of $L(t)$ is not necessarily coercive or bounded from above and $W(t,x)$ is not necessarily integrable on $\mathbb{R}$ with respect to $t$. Our theorem generalizes many known results in the references.

In this paper, we obtain the multiplicity of homoclinic solutions for a class of asymptotically autonomous Hamiltonian systems with indefinite sign potentials. The concentration-compactness principle is applied to show the compactness. As a byproduct, we obtain the uniqueness of the positive ground state solution for a class of autonomous Hamiltonian systems and the best constant for Sobolev inequality which are of independent interests.

In this paper, we study the existence of multiple homoclinic solutions for the following second order Hamiltonian systems \begin{equation*} \ddot{u}(t)-L(t)u(t)+\nabla W(t,u(t))=0, \end{equation*} where $L(t)$ satisfies a boundedness assumption which is different from the coercive condition and $W$ is a combination of subquadratic and superquadratic terms.

In this paper, the existence of non-constant periodic solutions for a class of conservative Hamiltonian systems with prescribed energy is obtained by the saddle point theorem.