On the interplay between $(p,q)$-growth and $x$-dependence of the energy integrand: a limit case

We establish the local Lipschitz regularity of the local minimizers of non autonomous integral funtionals of the form $${\int }_{\mathrm{\Omega }}F(x,Dz)dx,$$ where $\mathrm{\Omega }$ is a bounded open set of ${\mathbb{R}}^n$, $n\ge 2$. The energy density $F(x,\xi )$ satisfies $(p,q)-$growth conditions with respect to the gradient variable and belongs to the Sobolev class $W^{1,\phi }$, with $\phi (t)=t^r{\log }^{\alpha }(e+t),$ $r\ge n$, $\alpha \ge 0$, as a function of the $x$ variable, under the condition $$1\le \frac{q}{p}\le 1+\frac{1}{n}-\frac{1}{r}\mathrm{.}$$ We present a unified approach that covers the limit case $$\frac{q}{p}=1+\frac{1}{n}-\frac{1}{r}$$ and retrieves the results in \cite{EMM16} and in \cite{CGHPdN20}.