Pairs of positive solutions of a quasilinear elliptic Neumann problem driven by the mean curvature operator

We establish the existence of multiple positive weak solutions of the quasilinear elliptic Neumann problem driven by the mean curvature operator (formula Presented) 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.