Splitting-type variational problems with asymmetrical growth conditions

Splitting-type variational problems $${\int }_{\mathrm{\Omega }}\sum _{i=1}^n{f}_i({\mathrm{\partial }}_{i}w)dx\to \min$$ with superlinear growth conditions are studied by assuming $$h_i(t)\le f_i^{\mathrm{\prime }\mathrm{\prime }}(t)\le H_i(t)$$ with suitable functions $h_i$, $H_i$: $\mathbb{R}\to {\mathbb{R}}^{+}$, $i=1$, $\dots$ , $n$, measuring the growth and ellipticity of the energy density. Here, as the main feature, a symmetric behaviour like $h_i(t)\approx h_i(-t)$ and $H_i(t)\approx H_i(-t)$ for large $\mathrm{\mid }t\mathrm{\mid }$ is not supposed. Assuming quite weak hypotheses as above, we establish higher integrability of $\mathrm{\mid }\mathrm{\nabla }u\mathrm{\mid }$ for local minimizers $u\in L^{\mathrm{\infty }}(\mathrm{\Omega })$ by using a Caccioppoli-type inequality with some power weights of negative exponent.