The periodic problem for curvature-like equations with asymmetric perturbations

We discuss existence and multiplicity of solutions of the periodic problem for the curvature-like equation\begin{equation*}-\Big( u'/{ \sqrt{a^2+{u'}^2}}\Big)' = f(t,u) \end{equation*} by means of variational techniques in the space of bounded variation functions. As $a= 0$ is allowed,both the prescribed curvature equation and the $1$-Laplace equation are considered. We are concerned with the case where the right-hand side $f$ of the equation interacts with the beginning of the spectrum of the $1$-Laplace operator with periodic boundary conditions on $[0,T]$, being mainly interested in the situation where $\supess{[0,T]\times\RR}{f(t,s)} $ may differ from $-\infess{[0,T]\times \RR}{f(t,s)}$.