We establish the existence of multiple positive weak solutions of the quasilinear elliptic Neumann problem driven by the mean curvature operator ( −div ∇u/ p 1 + |∇u|2 _ = λw(x) |u|p−2u in Ω, −∇u ν/ p 1 + |∇u|2 = 0 on ∂Ω. Here, Ω is a bounded regular domain in RN, with N ≥ 2, p ∈ (1, 1∗), w is a sign-changing weight function, and λ > 0 is a parameter. Our findings provide the existence, for sufficiently small λ, of two positive solutions, the smaller in C1(Ω), the larger in BV (Ω), which respectively bifurcate from (λ, u) = (0, 0) and from (λ, u) = (0,+∞). This way we extend to a genuine PDE setting some results obtained in [22, 23] for the corresponding one-dimensional problem.
