  We study weak solutions and minimizers $u$ of the non-autonomous problems
$\operatorname{div} A(x, Du)=0$ and $\min_v \int_\Omega F(x,Dv)\,dx$ with
quasi-isotropic $(p, q)$-growth. We consider the case that $u$ is bounded,
H\"older continuous or lies in a Lebesgue space and establish a sharp
connection between assumptions on $A$ or $F$ and the corresponding norm of $u$.
We prove a Sobolev--Poincar\'e inequality, higher integrability and the
H\"older continuity of $u$ and $Du$. Our proofs are optimized and streamlined
versions of earlier research that can more readily be further extended to other
settings.
  Connections between assumptions on $A$ or $F$ and assumptions on $u$ are
known for the double phase energy $F(x, \xi)=|\xi|^p + a(x)|\xi|^q$. We obtain
slightly better results even in this special case. Furthermore, we also cover
perturbed variable exponent, Orlicz variable exponent, degenerate double phase,
Orlicz double phase, triple phase, double variable exponent as well as variable
exponent double phase energies and the results are new in most of these special
cases.
