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

Abstract

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,$\{\begin{array}{cc}-\mathrm{div}(\nabla u/\sqrt{1+|\nabla u{|}^2})=\lambda g\left(x\right)h\left(u\right)& \text{ in }\Omega ,\\ u=\phi & \text{ on }{\partial }_D\Omega ,\\ -\nabla u{\nu }_{\partial \Omega }/\sqrt{1+|\nabla u{|}^2}=\psi & \text{ on }{\partial }_N\Omega .\end{array}$ Here, Ω is a bounded domain in $R^N$ with a $C^{0,1}$ boundary ∂Ω and unit outer normal ${\nu }_{\Omega }$ such that $\partial \Omega ={\partial }_D\Omega \cup {\partial }_N\Omega$, ${\partial }_D\Omega \ne \varnothing$, and ${\partial }_D\Omega \cap {\partial }_N\Omega =\varnothing$, $g\in L^N\left(\Omega \right)$, $h\in C^0\left(\right[0,+\infty \left)\right)$, $\phi \in L^1(\partial {\Omega }_D)$, $-\psi \in L^{\infty }(\partial {\Omega }_N)$ 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.