Behaviour of solutions to p-Laplacian with Robin boundary conditions as p goes to 1

We study the asymptotic behaviour, as p→1+, of the solutions of the following inhomogeneous Robin boundary value problem: −Δpup=f in Ω, |∇up|p−2∇up⋅ν+λ|up|p−2up=g on ∂Ω, where Ω is a bounded domain in RN with sufficiently smooth boundary, ν is its unit outward normal vector and Δpv is the p-Laplacian operator with p>1. The data f∈LN,∞(Ω) (which denotes the Marcinkiewicz space) and λ,g are bounded functions defined on ∂Ω with λ≥0. We find the threshold below which the family of p–solutions goes to 0 and above which this family blows up. As a second interest we deal with the 1-Laplacian problem formally arising by taking p→1+ in (P).