Splitting-type variational problems ∫Ω∑i=1nfi(∂iw)dx→minwith superlinear growth conditions are studied by assuming hi(t)≤fi′′(t)≤Hi(t)(∗)with suitable functions hi, Hi: R→R+, i=1, ..., n, measuring the growth and ellipticity of the energy density. Here, as the main feature, we do not impose a symmetric behaviour like hi(t)≈hi(-t)and Hi(t)≈Hi(-t)for large |t|. Assuming quite weak hypotheses on the functions appearing in (∗), we establish higher integrability of |∇u|for local minimizers u∈L∞(Ω)by using a Caccioppoli-type inequality with some power weights of negative exponent.