Regularity results for a class of obstacle problems with p, q-growth conditions

In this paper we prove the the local Lipschitz continuity for solutions to a class of obstacle problems of the type(formula presented) Here K () is the set of admissible functions z 2 u0 +W1;p() for a given u0 2 W1;p() such that z a.e. in , being the obstacle and being an open bounded set of Rn, n 2. The main novelty here is that we are assuming that the integrand F(x;Dz) satises (p; q)-growth conditions and as a function of the x-variable belongs to a suitable Sobolev class. We remark that the Lipschitz continuity result is obtained under a sharp closeness condition between the growth and the ellipticity exponents. Moreover, we impose less restrictive assumptions on the obstacle with respect to the previous regularity results. Furthermore, assuming the obstacle is locally bounded, we prove the local boundedness of the solutions to a quite large class of variational inequalities whose principal part satisfies non standard growth conditions.