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

We study a class of semi-linear mean curvature equations −Mu = H + λf(u) where M is the mean curvature operator. 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∗. In the radial case, we then show that minimal solutions are classical solutions for all λ ∈ [0, λ∗] and that another branch of solutions exists in a neighborhood [λ∗ − η, λ∗] of λ∗. 2000 Mathematics Subject Classification: 49F10 (35J60)