We study autonomous integrals F[u] := ∫Ω f(Du) dx for u: ℝn ⊃ Ω → ℝN in the multidimensional calculus of variations, where the integrand f is a strictly quasiconvex function satisfying the (p, q)-growth conditions γ|ξ|p ≤ f(ξ) ≤ Γ(1+|ξ|q) with exponents 1 < p ≤ q < p+min{2, p}/(2n). Imposing the additional assumption that f resembles the degenerate behavior of the p-energy density, we establish a partial C1,α-regularity theorem for F-minimizers and a similar theorem for minimizers of a relaxed functional. Our results cover the model case of polyconvex integrands f(ξ) := 1/p |ξ|p + h(det ξ), where h is a smooth convex function with q/n-growth.