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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 By introducing a new superquadratic condition, we obtain the existence of two nontrivial homoclinic solutions for a class of perturbed second order Hamiltonian systems which are obtained by the mountain pass theorem and Ekeland’s variational principle.

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, we study the existence of homoclinic solutions to the following second-order Hamiltonian systems ü(t)-L(t)u(t)+▾ W(t,u(t))=0,∀t∊R, where L(t) is a symmetric and positive definite matrix for all t ∊ R. The nonlinear potential W is a combination of superlinear and sublinear terms. By different conditions on the superlinear and sublinear terms, we obtain existence and nonuniqueness of nontrivial homoclinic solutions to above systems. [ABSTRACT FROM AUTHOR]

When the nonlinearities are sublinear or asymptotically linear at infinity, we obtain the existence of negative energy solutions for a class of non-periodic discrete Schrödinger systems with perturbed terms. Besides, for the asymptotically linear case, we obtain the existence of positive energy solutions. We also illustrate our results by giving several examples. [ABSTRACT FROM AUTHOR]