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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. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of multiple homoclinic solutions for the following second order Hamiltonian systems ü(t) - L(t)u(t) + ∇W(t, u(t)) = 0, where L(t) satisfies a boundedness assumption which is different from the coercive condition and W is a combination of subquadratic and superquadratic terms. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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.