Nonexistence, existence and multiplicity of positive solutions to the p(x)-Laplacian nonlinear Neumann boundary value problem

We study the nonexistence, existence and multiplicity of positive solutions for the nonlinear Neumann boundary value problem involving the p(x)-Laplacian of the form {-Delta(p(x))u + lambda vertical bar u vertical bar(p(x)-2)u = f (x, u) in Omega vertical bar del vertical bar(p(x)-2)partial derivative u/partial derivative eta = g(x, u) on partial derivative Omega, where Omega is a bounded smooth domain in R-N, p is an element of C-1 ((Omega) over bar) and p(x) > 1 for x is an element of (Omega) over bar. Using the sub-supersolution method and the variational principles, under appropriate assumptions on f and g, we prove that there exists lambda(*) > 0 such that the problem has at least two positive solutions if lambda > lambda(*), has at least one positive solution if lambda = lambda(*) and has no positive solution if lambda < lambda(*).