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 { −Δpu+ a(x)up−1 = −u in Ω, |∇u|p−2 ∂u ∂ν = λup−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 → ∞.