On a class of obstacle problems with (p, q)-growth and explicit u-dependence

In this paper, we establish the higher differentiability of the gradient of solutions to variational obstacle problems of the type min (Formula presented). Here ω c ℝ n is a bounded open set, ψ in W 1,q0(Ω) is a fixed function called obstacle, and ψ (ω) Kψ(Ω) is the class of admissible functions. The main feature of the energy densities under consideration here is that they satisfy non-standard growth conditions with respect to the gradient variable and that they explicitly depend on the pair (x,u). Assuming that ψ L loc ∞ (ω) W loc 2, 2q - p (ω) ψ L∞ loc(Ω), we are able to prove a second order regularity result for the solution.