Limit cases in an elliptic problem with a parameter-dependent boundary condition

In this work we discuss existence, uniqueness and asymptotic profiles of positive solutions to the quasilinear problem

 −Δ_pu+a(x)u^p−1=−u^r in Ω,

 |∇u|^p−2 ∂u/∂ν=λu^p−1 on ∂Ω

for λ∈R, where r>p−1>0, a∈L^∞(Ω). We analyze the existence of solutions in terms of a principal eigenvalue, and determine their asymptotic behavior both when r→p−1 and when r→∞.