Existence and multiplicity results for elliptic problems with p(.)-Growth conditions

The variable exponent spaces are essential in the study of certain nonhomogeneous materials. In the framework of these spaces, we are concerned with a nonlinear elliptic problem involving a p(.)-Laplace-type operator on a bounded domain ohm subset of R-N (N >= 2) of smooth boundary partial derivative ohm. We introduce the variable exponent Sobolev space of the functions that are constant on the boundary and we show that it is a separable and reflexive Banach space. This is the space where we search for weak solutions to our equation -div(a(x, del u)) + vertical bar u vertical bar(p(x)-2)u = lambda f (x, u), provided that lambda >= 0 and a : (ohm) over bar x R-N -> R-N, f : ohm x R -> R are fulfilling appropriate conditions. We use different types of mountain pass theorems, a classical Weierstrass type theorem and several three critical points theorems to establish existence and multiplicity results under different hypotheses. We treat separately the case when f has a p(.) - 1-superlinear growth at infinity and the case when f has a p(.) - 1-sublinear growth at infinity. (c) 2012 Elsevier Ltd. All rights reserved.