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

We study a class of mean curvature equations where 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*.