Any interesting literature with interesting problems on a generalized Laplacian?

I have the notion of a generalized laplacian such as this: $\nabla^2=\sum_{i=1}^n \frac{\partial^2}{\partial x_i^2}$

So suppose I look at the generalized laplacian defined as such: $\nabla^{2m}=(\nabla^2)^m$

where $m\in \mathbb{Z}$. How weould one solve $\nabla^{2m}u=0$ or when the RHS isn't necessarily zero.

Any interesting literature on such a notion of PDEs?