Homoclinic solutions for second-order Hamiltonian systems with periodic potential

Abstract

Abstract In this paper, we study the second-order Hamiltonian systems u¨−L(t)u+∇W(t,u)=0,t∈R, $$ \ddot{u}-L(t)u+\nabla W(t,u)=0,\quad t\in \mathbb{R}, $$ where L∈C(R,RN×N) $L\in C(\mathbb{R},\mathbb{R}^{N\times N})$ is a T-periodic and positive definite matrix for all t∈R $t\in \mathbb{R}$ and W is superquadratic but does not satisfy the usual Ambrosetti–Rabinowitz condition at infinity. One ground homoclinic solution is obtained by applying the monotonicity trick of Jeanjean and the concentration–compactness principle. The main result improves the recent result of Liu–Guo–Zhang (Nonlinear Anal., Real World Appl. 36:116–138, 2017).

Keywords

• Homoclinic solution
• Ground state
• Variational methods
• Vanishing
• Nonvanishing