Positive solutions of the prescribed mean curvature equation with non-homogeneous mixed boundary conditions

We investigate existence, non-existence, multiplicity, stability, and regularity issues for the positive bounded variation solutions of the prescribed mean curvature equation with non-zero mixed, Dirichlet-Neumann, boundary data, \begin{equation} \left\{ \begin{array}{clll} \displaystyle -{\rm div } \Big( {\nabla u}/{ \sqrt{1+|\nabla u|^2}}\Big) = \la g(x) h(u) & \displaystyle \hbox{\, in $\Omega$}, \\[1mm] \displaystyle u=\v & \displaystyle \hbox{\, on $\partial_\DD\Omega$}, \\[1mm] \displaystyle -\nabla u \, \nu_{\partial \Omega} / \sqrt{1+|\nabla u|^2} =\psi & \displaystyle \hbox{\, on $\partial_\NNN\Omega$}. \end{array} \right. \end{equation} Here, $\Omega$ is a bounded domain in ${\mathbb R}^N$ with a $C^{0,1}$ boundary $\partial \Omega$ and unit outer normal $\nu_\Omega$ such that $\partial \Omega =\partial_\DD\Omega \cup \partial_\NNN\Omega$, $\partial_\DD\Omega\neq \emptyset$, and $\partial_\DD\Omega \cap \partial_\NNN\Omega = \emptyset$, $g\in L^N(\Omega)$, $h \in C^0([0,+\infty) ) $, $\v\in L^1(\partial \Omega_\DD)$, $-\psi\in L^\infty(\partial \Omega_\NNN)$ are positive functions, and $\lambda\in [0,+\infty) $ is a parameter. All results, about existence and multiplicity in particular, are obtained without growth restritions on the function $h$ being imposed.