On the local everywhere Hölder continuity of the minima of a class of vectorial integral functionals of the calculus of variations

In this paper we study the everywhere Hölder continuity of the minima of the following class of vectorial integral funcionals:
$$ \int_{\Omega}\sum_{\alpha=1}^{m}|\nabla u^{\alpha}|^{p}+G\bigl(x,u,|\nabla u^{1}|,\dots,|\nabla u^{m}|\bigr) \,dx, $$
with some general conditions on the density $G$.
We make the following assumptions about the function $G$. Let $\Omega$ be a bounded open subset of $\mathbb{R}^{n}$, with $n\geqslant 2$, and let $G \colon \Omega \times\mathbb{R}^{m}\times\mathbb{R}_{0,+}^{m}\to\mathbb{R}$ be a Carathéodory function, where $\mathbb{R}_{0,+}=[0,+\infty)$ and $\mathbb{R}_{0,+}^{m}=\mathbb{R}_{0,+}\times \dots \times\mathbb{R}_{0,+}$ with $m\geqslant 1$. We make the following growth conditions on $G$: there exists a constant $L>1$ such that
\begin{align*} \sum_{\alpha=1}^{m}|\xi^{\alpha}|^{q}-\sum_{\alpha=1}^{m}|s^{\alpha}|^{q}-a(x) &\leqslant G\bigl(x,s^{1},\dots,s^{m},|\xi^{1}|,\dots,|\xi^{m}|\bigr) \\ &\leqslant L\biggl[\sum_{\alpha=1}^{m}|\xi^{\alpha}|^{q}+\sum_{\alpha=1}^{m}|s^{\alpha}|^{q}+a(x) \biggr] \end{align*}
for $\mathcal{L}^{n}$ a.e. $x\in \Omega $, for every $s^{\alpha}\in\mathbb{R}$ and every $\xi^{\alpha}\in\mathbb{R}$ with $\alpha=1,\dots,m$, $m\geqslant 1$ and with $a(x) \in L^{\sigma}(\Omega)$, $a(x)\geqslant 0$ for $\mathcal{L}^{n}$ a.e. $x\in \Omega$, $\sigma >{n}/{p}$, $1\leqslant q<p^2/n$ and $1<p<n$.
Assuming that the previous growth hypothesis holds, we prove the following regularity result. If $u\in W^{1,p}(\Omega,\mathbb{R}^{m})$ is a local minimizer of the previous functional, then $u^{\alpha}\in C_{\mathrm{loc}}^{o,\beta_{0}}(\Omega) $ for every $\alpha=1,\dots,m$, with $\beta_{0}\in (0,1) $. The regularity of minimizers is obtained by proving that each component stays in a suitable De Giorgi class and, from this, we conclude Hölder continuity.