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.
