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

We prove regularity results for minimizers of functionals F(u,Ω) := Ω f (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 type L−1|z|p(x) f (x, ξ, z) L 1 + |z|p(x) , with growth exponent p :Ω →(1,∞).