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