Infinitely many fast homoclinic solutions for a class of superquadratic damped vibration systems

Abstract

Consider the following damped vibration system $$\begin{aligned} \ddot{u}(t)+q(t)\dot{u}(t)-L(t)u(t)+\nabla W(t,u(t))=0,\ \forall t\in \mathbb {R} \qquad \qquad (1) \end{aligned}$$ where $$q\in C(\mathbb {R},\mathbb {R})$$ , $$L\in C(\mathbb {R},\mathbb {R}^{N^{2}})$$ and $$W\in C(\mathbb {R}\times \mathbb {R}^{N},\ \mathbb {R})$$ . Applying a Symmetric Mountain Pass Theorem, we prove the existence of infinitely many fast homoclinic solutions for (1) when L is not required to be either uniformly positive definite or coercive and W satisfies some general super-quadratic conditions at infinity in the second variable but does not satisfy the classical superquadratic growth conditions at infinity.