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

Consider Robin problem involving the p(x)-Laplacian on a smooth bounded domain Omega as follows {-Delta(rho(x))u =lambda f (x, u) in Omega, vertical bar del u vertical bar(rho(x)-2)partial derivative u/partial derivative eta + beta vertical bar u vertical bar(rho(x)-2)u = 0 on partial derivative Omega. Applying the sub-supersolution method and the variational method, under appropriate assumptions on f, we prove that there exists lambda(*) > 0 such that the problem has at least two positive solutions if lambda is an element of (0, lambda(*)), has at least one positive solution if lambda = lambda(*) < + infinity and has no positive solution if lambda > lambda(*). To prove the results, we prove a norm on W-1.rho(x)(Omega) without the part of vertical bar .vertical bar L-rho(x)( Omega) which is equivalent to usual one and establish a special strong comparison principle for Robin problem. (C) 2009 Elsevier Inc. All rights reserved.