Infinitely Many Fast Homoclinic Solutions for Damped Vibration Systems with Combined Nonlinearities

This article  concerns the existence of fast homoclinic solutions for the following damped vibration system\begin{equation*}\frac{d}{dt}(P(t)\dot{u}(t))+q(t)P(t)\dot{u}(t)-L(t)u(t)+\nabla W(t,u(t))=۰,\end{equation*}where P,L\in C\left(\mathbb{R},\mathbb{R}^{N^{۲}}\right) are symmetric and positive definite matrices, q\in C\left(\mathbb{R},\mathbb{R}\right) and W\in C^{۱}\left(\mathbb{R}\times\mathbb{R}^{N},\mathbb{R}\right). Applying the Fountain Theorem and Dual Fountain Theorem, we prove the above system possesses two different sequences of fast homoclinic solutions when L satisfies a new coercive condition and the potential W(t,x) is combined nonlinearity.

کلیدواژه ها:

Damped vibration systems ، Fast homoclinic solutions ، Variational methods ، Fountain Theorem ، Dual Fountain Theorem