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

We study a class of mean curvature equations −Mu=H+λup where M denotes the mean curvature operator and for p⩾1. We show that there exists an extremal parameter λ∗ such that this equation admits a minimal weak solutions for all λ∈[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∈[0,λ∗] and that another branch of classical solutions exists in a neighborhood (λ∗−η,λ∗) of λ∗.