Existence and regularity of extremal solutions for a mean-curvature equation

Abstract

We study a class of mean curvature equations $-\mathcal{M}u=H+\lambda u^p$ where $\mathcal{M}$ denotes the mean curvature operator and for $p⩾1$. We show that there exists an extremal parameter ${\lambda }^{\ast }$ such that this equation admits a minimal weak solutions for all $\lambda \in [0,{\lambda }^{\ast }]$, while no weak solutions exists for $\lambda >{\lambda }^{\ast }$ (weak solutions will be defined as critical points of a suitable functional). In the radially symmetric case, we then show that minimal weak solutions are classical solutions for all $\lambda \in [0,{\lambda }^{\ast }]$ and that another branch of classical solutions exists in a neighborhood $({\lambda }_{\ast }-\eta ,{\lambda }^{\ast })$ of ${\lambda }^{\ast }$.