Positive solutions for Robin problem involving the -Laplacian

Consider Robin problem involving the p ( x ) -Laplacian on a smooth bounded domain Ω as follows { − Δ p ( x ) u = λ f ( x , u ) in Ω , | ∇ u | p ( x ) − 2 ∂ u ∂ η + β | u | p ( x ) − 2 u = 0 on ∂ Ω . Applying the sub-supersolution method and the variational method, under appropriate assumptions on f, we prove that there exists λ * > 0 such that the problem has at least two positive solutions if λ ∈ ( 0 , λ * ) , has at least one positive solution if λ = λ * < + ∞ and has no positive solution if λ > λ * . To prove the results, we prove a norm on W 1 , p ( x ) ( Ω ) without the part of | ⋅ | L p ( x ) ( Ω ) which is equivalent to usual one and establish a special strong comparison principle for Robin problem.