= Solution
For the <p-energy>
$$
F[u]=\frac1p\int_\Omega|Du|^p\,dx,
$$
the <first variation> in the direction $\psi\in C_c^\infty(\Omega)$ is
$$
\left.\frac d{dt}F[u+t\psi]\right|_{t=0}
=\int_\Omega|Du|^{p-2}Du\mathbin\cdot D\psi.
$$
An <integration by parts> therefore gives the <Euler-Lagrange equation>
$$
\operatorname{div}(|Du|^{p-2}Du)=0,
$$
which is the <p-Laplacian equation>. In the notation of the question one takes $a^{ij}(x,z,\eta)=|\eta|^{p-2}\delta^{ij}$.
For $\eta\ne0$, the principal coefficient matrix is
$$
A^{ik}(\eta)=|\eta|^{p-2}\delta^{ik}
+(p-2)|\eta|^{p-4}\eta_i\eta_k.
$$
Its eigenvalue in directions orthogonal to $\eta$ is $|\eta|^{p-2}$, while its eigenvalue parallel to $\eta$ is $(p-1)|\eta|^{p-2}$. The coefficients are $C^{1,\alpha}$ away from $\eta=0$, and the <condition number> there is at most $p-1$. On every region where $0<m\leq|Du|\leq M$, this gives uniform ellipticity with constants depending on $m$, $M$, and $p$. At $Du=0$ all principal eigenvalues vanish, so the operator is degenerate there and is not strictly elliptic on a domain containing a critical point.
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