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

Abstract

We study a class of mean curvature equations $-\mathcal Mu=H+\lambda u^p$ where $\mathcal M$ denotes the mean curvature operator and for $p\geq 1$. We show that there exists an extremal parameter $\lambda^*$ such that this equation admits a minimal weak solutions for all $\lambda \in [0,\lambda^*]$, while no weak solutions exists for $\lambda >\lambda^*$ (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 $\lambda\in [0,\lambda^*]$ and that another branch of classical solutions exists in a neighborhood $(\lambda_*-\eta,\lambda^*)$ of $\lambda^*$.