Regularity results for a class of obstacle problems under non standard 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, ∞).