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.

How to Cite this Article:

Mohsen Timoumi, Existence and multiplicity of fast homoclinic solutions for a class of damped vibration problems, Journal of Nonlinear Functional Analysis 2016 (2016), Article ID 9.