Infinitely Many Solutions for Fractional Hamiltonian Systems with Locally Defined Potentials

This article concerns the existence of fast homoclinic solutions for the following damped vibration system where $P,L\in C(\mathbb{R},\mathbb{R}^{N^{2}})$ are symmetric and positive definite matrices, $q\in C(\mathbb{R},\mathbb{R})$ and $W\in C^{1}(\mathbb{R}\times\mathbb{R}^{N},\mathbb{R})$. Applying Fountain Theorem and Dual Fountain Theorem, we prove that system (1) possesses two different sequences of fast homoclinic solutions when $L$ satisfies a new coercive condition and the potential $W(t,x)$ is with combined nonlinearities.