Existence and non-existence results for a quasilinear problem with nonlinear boundary condition

Abstract We study the problem − div a x |∇ u | p − 2 ∇ u = λ 1 + | x | α 1 |u| q − 2 u − h x |u| r − 2 u in Ω ⊂ R N , a x |∇u| p − 2 ∇u · n + b x · |u| p − 2 u = θg x , u on Γ, u ≥ 0 in Ω, where Ω is an unbounded domain with smooth boundary Γ, n denotes the unit outward normal vector on Γ, and λ > 0, θ are real parameters. We assume throughout that p q r p* = pN N − p , 1 p N, −N α 1 q · N − p p − N , while a , b , and h are positive functions. We show that there exist an open interval I and λ* > 0 such that the problem has no solution if θ ∈  I and λ ∈ (0, λ*). Furthermore, there exist an open interval J  ⊂  I and λ 0  > 0 such that, for any θ ∈  J , the above problem has at least a solution if λ ≥ λ 0 , but it has no solution provided that λ ∈ (0, λ 0 ). Our paper extends previous results obtained by J. Chabrowski and K. Pfluger.