Existence, regularity and stability properties of periodic solutions of a capillarity equation in the presence of lower and upper solutions

We develop a lower and upper solutions method for the periodic problem associated with the capillarity equation \begin{equation*} -\Big( u'/{ \sqrt{1+{u'}^2}}\Big)' = f(t,u) \end{equation*} in the space of bounded variation functions. We get the existence of periodic solutions both in the case where the lower solution $\alpha$ and the upper solution $\beta$ satisfy $\alpha \le \beta$, and in the case where $\alpha \not\le \beta$. In the former case we also prove regularity and order stability of solutions.