Mean oscillation conditions for nonlinear equation and regularity results

We consider general nonlinear elliptic equations of the form $$\mathrm{div}A(x,Du)=0\text{in }\mathrm{\Omega },$$ where $A:\mathrm{\Omega }\times {\mathbb{R}}^n\to {\mathbb{R}}^n$ satisfies a quasi-isotropic $(p,q)$-growth condition, which is equivalent to the point-wise uniform ellipticity of $A$. We establish sharp and comprehensive mean oscillation conditions on $A(x,\xi )$ with respect to the $x$ variable to obtain $C^1$- and $W^{1,s}$-regularity results. The results provide new conditions even in the standard $p$-growth case with coefficient $\mathrm{div}(a(x)\mathrm{\mid }Du{\mathrm{\mid }}^{p-2}Du)=0$. Also included are variable exponent growth with and without perturbation as well as borderline double-phase growth and double-phase growth with a coefficient.