I'm currently working on this Dirichlet problem:

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

with $\sigma \in L^{\infty}_{+}(\Omega)$, boundary $g \in W^{1,p}(\Omega)$ and $f \in L^2(\Omega)$.

I think I've already proved that this problem has a unique solution, but I don't know how to prove the following a priori estimate

\begin{align} ||{u}||_{W^{1,p}} \leq C ({||g||}_{W^{1,p}} + {||f||}_{L^2}^{\frac{1}{p-1}}) \end{align}

This estimate is mentioned in this paper https://pdfs.semanticscholar.org/7399/da07c625d51aa7ee72840789916b036019d2.pdf (second page; equation (1.4)) but without giving proof. I guess it's seen "easy", but I just don't know how to get there. I know how to conclude an a priori estimate in the case $f=0$ or $g=0$, but since p-Laplace operator isn't linear, this doesn't help and I'm really clueless know. I hope someone can help me with with.

As far as i thought i can use the Poincaré inequality since $u - g \in W^{1,p}_0$:

\begin{align} ||u||_{W^{1,p}} &= ||u||_{L^p} + ||\nabla u||_{L^p}\\ & \leq ||g||_{L^p} + ||u - g||_{L^p} + ||\nabla u||_{L^p}\\ & \leq ||g||_{L^p} + C||\nabla(u - g)||_{L^p} + ||\nabla u||_{L^p} \end{align}

The problem is that i can't find an upper bound for $||\nabla u||_{L^p}$ using $f$...

The inequality in the paper that i found implies that i should found something like

\begin{align} ||\nabla u||^p_{L^p} \leq ||f||_{L^2}~||\nabla u||_{L^p} \end{align}

which only be valid, when the solution u is in $W^{1,p}_0$...

I hope someone can help me with with this...