| Contents |
| We consider the periodic problem associated with the capillarity equation
$$
\displaystyle
-\Big( u'/{ \sqrt{1+{u'}^2}}\Big)'
= f(t,u).
$$ In the setting of bounded variation functions we present some existence results in the presence of lower and upper 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 find a solution as a local minimizer of the associated action functional without any additional assumption on the right-hand side $f$. In the latter case, we still prove the existence of at least one solution, but now a control on $f$ is needed with respect to the first branch of the Dancer-Fu$\check{\mbox{c}}$ik spectrum of the $T$-periodic problem for the $1$-Laplace operator. Some results concerning regularity and stability are also produced. |
|