Positive solutions for Robin problem involving the p(x)p(x)-Laplacian

Consider Robin problem involving the p(x)p(x)-Laplacian on a smooth bounded domain Ω as follows{−Δp(x)u=λf(x,u)in Ω,|∇u|p(x)−2∂u∂η+β|u|p(x)−2u=0on ∂Ω. Applying the sub-supersolution method and the variational method, under appropriate assumptions on f  , we prove that there exists λ*>0λ*>0 such that the problem has at least two positive solutions if λ∈(0,λ*)λ∈(0,λ*), has at least one positive solution if λ=λ*<+∞λ=λ*<+∞ and has no positive solution if λ>λ*λ>λ*. To prove the results, we prove a norm on W1,p(x)(Ω)W1,p(x)(Ω) without the part of |⋅|Lp(x)(Ω)|⋅|Lp(x)(Ω) which is equivalent to usual one and establish a special strong comparison principle for Robin problem.