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

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

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