Existence and multiplicity results for the prescribed mean curvature equation via lower and upper solutions

We discuss existence and multiplicity of bounded variation solutions of the mixed problem for the prescribed mean curvature equation$$-{\rm div } \Big({\nabla u}/{ \sqrt{1+{|\nabla u|}^2}}\Big) = f(x,u) \hbox{\, in $\Omega$},\quadu=0 \hbox{\, on $\Gamma_{D}$}, \quad \partial u / \partial \nu =0 \hbox{\, on $ \Gamma_{N}$}, $$where $\Gamma_{D} $ is an open subset of $\partial \Omega$ and $\Gamma_{N}=\partial \Omega\setminus \Gamma_{D}$. Our approach is based on variational techniques and a lower and upper solutions method specially developed for this problem.