Multiplicity of Positive Solutions for a Class of Inhomogeneous Neumann Problems Involving the p(x)-Laplacian

We study the existence and multiplicity of positive solutions for the inhomogeneous Neumann boundary value problems involving the p(x)-Laplacian of the form where Ω is a bounded smooth domain in , and p(x) > 1 for with and φ ≢ 0 on ∂Ω. Using 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 = λ_*, has at least one positive solution if λ = λ_*, and has no positive solution if λ = λ_*. To prove the result we establish a special strong comparison principle for the Neumann problems. The research was supported by the National Natural Science Foundation of China 10371052,10671084).