On locally superquadratic 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 , $$ \ddot{u}-L(t)u+\nabla W(t,u)=0, $$ where t ∈ R $t\in \mathbb{R}$ , u ∈ R N $u\in \mathbb{R}^{N}$ , L and W depend periodically on t, 0 lies in a spectral gap of the operator − d 2 / d t 2 + L ( t ) $-d^{2}/dt^{2}+L(t)$ and W ( t , x ) $W(t,x)$ is locally superquadratic. Replacing the common superquadratic condition that lim | x | → ∞ W ( t , x ) | x | 2 = + ∞ $\lim_{|x|\rightarrow \infty }\frac{W(t,x)}{|x|^{2}}=+\infty $ uniformly in t ∈ R $t\in \mathbb{R}$ by the local condition that lim | x | → ∞ W ( t , x ) | x | 2 = + ∞ $\lim_{|x|\rightarrow \infty }\frac{W(t,x)}{|x|^{2}}=+\infty $ a.e. t ∈ J $t\in J$ for some open interval J ⊂ R $J\subset \mathbb{R}$ , we prove the existence of one nontrivial homoclinic soluiton for the above problem.

Keywords

• Homoclinic solutions
• Hamiltonian systems
• Strongly indefinite functional
• Locally superquadratic
• Linking theorem