Regularity results for a class of obstacle problems under nonstandard growth conditions

We prove regularity results for minimizers of functionals F(u,Ω):=∫Ωf(x,u,Du)dx in the class K:={u∈W1,p(x)(Ω,R):u⩾ψ}, where ψ:Ω→R is a fixed function and f is quasiconvex and fulfills a growth condition of the typeL−1|z|p(x)⩽f(x,ξ,z)⩽L(1+|z|p(x)), with growth exponent p:Ω→(1,∞).