It is well-known that if $O$ is an orthogonal map, then $\Delta u(Ox) = \Delta u$ where $\Delta$ is the Laplacian. 

Now, let $A$ be a constant invertible matrix, then we define the weighted Laplacian 

$$\Delta_A = \langle \nabla, A \nabla \rangle.$$

I wonder: Does there exist a function $\varphi_O$ such that $Gf(x):=e^{\varphi_O(x)}f(Ox)$ satisfies for all $f$ but fixed $O$

 
$$\Delta_A Gf(x) = G(\Delta_A f)(x)?$$

Please let me know if you have any questions.