characteristic equation

Aim. Development of a method of analytical research of a two-mass oscillating system with parallel elastic and damping elements, which makes it possible to expand the design of such systems in various tasks of the functioning of machines and equipment. Method. We will conduct a parametric study of the dynamic oscillation system to assess the effect of the elasticity coefficient and damping on the change in the natural frequency, using the Laplace transform method.

In this paper, we study the asymptotic behavior of radial solutions of the following quasi-linear equation with the Hardy potential $\Delta_p u+h(|x|)|u|^{p-2}u=0$, $x\in \mathbb{R}^{N}-\{0\}$, where $2<p<N$, $h$ is a radial function on $\mathbb{R}^{N}-\{0\}$ such that $h(|x|)=\gamma|x|^{-p}$, $\gamma>0$ and $\Delta_p u=\operatorname{div}\left(|\nabla u|^{p-2}\nabla u\right)$ is the $p$-Laplacian operator.  The study strongly depends on the sign of $\gamma-(\sigma/p^\ast)^p$ where $\sigma=(N-p)/(p-1)$ and $p^\ast=p/(p-1)$.