On an eigenvalue problem involving variable exponent growth conditions

We study the problem -Delta u - epsilon div((1 + vertical bar del u vertical bar(2))p(x)-2/2 del u) = lambda(u + epsilon) in Omega, u = 0 on partial derivative Omega, where Omega is a bounded domain in R-N, p (Omega) over bar -> (1, 2) is a continuous function and X and F are two positive constants. We prove that for any epsilon > 0 each) lambda is an element of (0, lambda(1)) is an eigenvalue of the above problem, where Omega is the principal eigenvalue of the Laplace operator on Q. Moreover, for each eigenvalue lambda is an element of (0, lambda(1)) it corresponds a unique eigenfunction. The proofs will be based on the Banach fixed point theorem combined with adequate variational techniques. (c) 2009 Elsevier Ltd. All rights reserved.