#### Mohsen Timoumi, Existence and multiplicity of fast homoclinic solutions for a class of damped vibration problems, 2016 (2016), Article ID 9 (17 March 2016)

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Abstract

In this paper, we investigate the existence and multiplicity of homoclinic solutions for the following damped vibration problems $\ddot{u}(t)+q(t)\dot{u}(t)-L(t)u(t)+\nabla W(t,u(t))=0,$ where $q:\mathcal{R}\longrightarrow\mathcal{R}$ is a continuous function, $L(t)\in C(\mathcal{R},\mathcal{R}^{n^{2}})$ is a symmetric matrix and $W(t,x)\in C^{1}(\mathcal{R}\times \mathcal{R}^{n},\mathcal{R})$ are neither autonomous nor periodic in $t$. The novelty of this paper is that, supposing that $\lim_{\left|t\right|\longrightarrow\infty}Q(t)=+\infty$ $(Q(t)=\int^{t}_{0}q(s)ds)$ and $L(t)$ is coercive unnecessary uniformly positively definite for all $t\in\mathcal{R}$, we establish one new compact embedding theorem. Subsequently, assuming $W(t,x)$ satisfies the super-quadratic condition $\frac{W(t,x)}{\left|x\right|^{2}}\longrightarrow+\infty$ as $\left|x\right|\longrightarrow\infty$ uniformly in $t$ and need not satisfy the global Ambrosetti-Rabinowitz condition, we obtain some new criterion to guarantee the existence and multiplicity of nontrivial homoclinic solutions for damped vibration problems using the Minimax Methods in critical point theory.