for all $\gamma>p_*-1$ and $\lambda<\lambda'\in{\left(}0,1/2{\right]}$, where $\delta_i:=1$ if $i\in\mathcal{I}_1$, $\delta_i:=2$ if $i\in\mathcal{I}_2$, $\gamma_i:=\frac{n-p}{n}\gamma+p_i-p$ for all $i\in\mathcal{I}_1\cup\mathcal{I}_2$, and $C=C{\left(}n,\operatorname{\overrightarrow{p}},{\overrightarrow{q}},u,\gamma{\right)}$ (see Lemma \[Lem4\] for more details on the dependence of the constant with respect to $u$ and $\gamma$). Since the right-hand side of involves different exponents $\gamma_i$ in the anisotropic case, the number of exponents in the estimates may grow exponentially when iterating this inequality. We overcome this issue in Section \[Sec5\] by controlling the values of the exponents with respect to the number of iterations.

In Section \[Sec6\], we prove a vanishing result which will give Point (i) in Theorem \[Th3\]. We prove this result by applying our iteration scheme with $R_1=R_2^{1/\varepsilon}$ for small real numbers $\varepsilon>0$ and $\mathcal{I}_1$, $\mathcal{I}_2$ being the sets of all indices $i$ such that $p_i<p_0$, $p_i=p_0$, respectively, for some large enough real number $p_0\in\operatorname{\overrightarrow{p}}$ (see for the exact condition on $p_0$). Passing to the limit into our iteration scheme, we obtain a pointwise estimate of the form $$\label{Eq8}
\left\|u\right\|_{L^\infty(\Omega_{{\overrightarrow{q}}}(\mathcal{I}_1,R^{1/\varepsilon},\mathcal{I}_2,R,1/4))}\le\big(CR^{-\frac{1}{p_\varepsilon}}\big)^{\frac{1}{\varepsilon}}$$ for some constant $C=C{\left(}n,\operatorname{\overrightarrow{p}},u{\right)}$ (see Lemma \[Lem8\]). When $R$ is large enough, the right-hand side of converges to 0 as $\varepsilon\to0$, and we thus obtain our vanishing result.

In Section \[Sec7\], we prove Theorems \[Th1\], \[Th2\], and 