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

We are concerned with the existence of multiple periodic solutions for differential equations involving Fisher-Kolmogorov perturbations of the relativistic operator of the form − [ ϕ ( u ′ ) ] ′ = λ u ( 1 − | u | q ) , \begin{equation*} -\left [\phi (u’)\right ]’=\lambda u(1-|u|^q), \end{equation*} as well as for difference equations, of type − Δ [ ϕ ( Δ u ( n − 1 ) ) ] = λ u ( n ) ( 1 − | u ( n ) | q ) ; \begin{equation*} -\Delta \left [\phi (\Delta u(n-1))\right ]=\lambda u(n)(1-|u(n)|^q); \end{equation*} here q > 0 q>0 is fixed, Δ \Delta is the forward difference operator, λ > 0 \lambda >0 is a real parameter and ϕ ( y ) = y 1 − y 2 ( y ∈ ( − 1 , 1 ) ) . \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 C^1 -functionals.