Finite element approximation of weighted p-laplacian - error estimation?

i'm currently working on the following dirichlet problmen:

\begin{cases} \text{div} (\sigma(x) |\nabla u|^{p-2} \nabla u) = f &\quad \text{in } \Omega\\ u = g &\quad \text{in } \partial\Omega \end{cases}

with $\sigma \in L^{\infty}_{+}(\Omega)$, $g \in W^{1 - 1/p}(\partial \Omega)$ ,$f \in L^2(\Omega)$ and $\Omega \subset R^2$ being a bounded open set

i'm trying to approx the solution with the finite element method and am looking for an estimate for the error

\begin{align} ||u - u_h||_{W^{1,p}(\Omega)} \end{align}

where $u$ is the solution and $u^h$ is approx FEM solution.

while searching the internet, i found the following error estimate for $\sigma(x) = 1$:

\begin{align} ||u - u_h||_{W^{1,p}(\Omega)} = \begin{cases} Ch^{p/2} \quad p\leq 2 \\ Ch^{2/p} \quad p \geq 2\end{cases} \end{align}

and then i found a paper of chow: https://link.springer.com/article/10.1007/BF01396320

he proves this error estimates for a more general version with:

\begin{cases} \text{div} (k(x, |\nabla u|) \nabla u) = f &\quad \text{in } \Omega\\ u = g &\quad \text{in } \partial\Omega \end{cases}

where the function $k(x, \cdot)$ has to fulfill the following conditions:

picture of the conditions

i think since $\sigma(x)$ is positive and bounded then

$k(x,t) = \sigma(x) t^{p-2}$

will fulfill these condition and the proven error estimate will be true for my specifi problem, or am i wrong because i am missing something?