Linearized operator of higher order $p$ Laplacian

First, there's maybe a more revealing way to write the linearization of the p-Laplacian: $$L_v(\phi) = -\text{div}(|Dv|^{p - 2}\mathscr{C} D\phi)$$ where we define the rank-4 tensor $$\mathscr{C} = I + (p - 2)\frac{Dv\otimes Dv}{|Dv|^2},$$ which is dimensionless. This is worth meditating on. I prefer writing it this way because it makes it obvious that the linearization is a divergence-form elliptic operator and is therefore symmetric and positive-definite.

To answer your question about generalizing the p-Laplace equation, I think the most helpful way to write the PDE is to instead pose it as minimizing the functional $$J(v) = \int_\Omega\left(\frac{1}{p}|Dv|^p - fv\right)dx.$$ But here we could replace $D$ by any differential operator $L$, for example the Laplace operator $\Delta$ or something of even higher order. In that case the Euler-Lagrange equations become $$L^*\left(|Lv|^{p - 2}Lv\right) = f$$ where $L^*$ is the adjoint of $L$. In the case where $L$ is the gradient, $L^*$ is $-\text{div}$. You may need other terms depending on boundary conditions and such.