Fisher-Kolmogorov type perturbations of the relativistic operator: differential vs. difference

Abstract:

We are concerned with the existence of multiple periodic solutions for differential equations involving Fisher-Kolmogorov perturbations of the relativistic operator of the form \begin{equation*} -\left [\phi (u’)\right ]’=\lambda u(1-|u|^q), \end{equation*} as well as for difference equations, of type \begin{equation*} -\Delta \left [\phi (\Delta u(n-1))\right ]=\lambda u(n)(1-|u(n)|^q); \end{equation*} here $q>0$ is fixed, $\Delta$ is the forward difference operator, $\lambda >0$ is a real parameter and \begin{equation*} \displaystyle \phi (y)=\frac {y}{\sqrt {1- y^2}}\quad (y\in (-1,1)). \end{equation*} The approach is variational and relies on critical point theory for convex, lower semicontinuous perturbations of $C^1$-functionals.