I have the original Poisson/Laplace equation: $ \nabla^2V = 0$, and I want to break that down (in the context of electrostatics in Physics), using $V(x,y) = I(x,y)\rho(x,y)$, where $x$, and $y$, are spatial coordinates, and $\rho(x,y)$ is resistivity.
I apply the chain rule and get the new PDE:
$\rho\nabla^2I + I\nabla^2\rho = 0$
I can input a functional form for $\rho(x,y)$ so that I can numerically solve for $I(x,y)$ under these conditions. My method of choice is using finite-differencing because it's the simplest method, and one that I am comfortable implementing.
Is there a well-known PDE that follows this form, and are there analytical solutions to it (e.g. for special cases)? While it doesn't seem impossible for an FD method to handle it, I want to have some sanity checks or assurance that the solution for $I(x,y)$ is unique, given some $\rho(x,y)$.
Any help would be appreciated! For context, it's been a while since my PDE class...