Positive solutions for Robin problem involving the -Laplacian

Consider Robin problem involving the p(x)-Laplacian on a smooth bounded domain Ω as follows

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.