In Evan's book is an example for finding radial solutions in polar coordinates. Now I'm searching for non constant solutions that satisfy the equation and depend only on $\varphi$.
$$\Delta_u = {\partial^2 u \over\partial r^2}+{1\over r}{\partial u\over\partial r}+{\partial^2 u \over\partial\varphi^2} =0$$
My guess is: $$u=v(\varphi)=c\cdot\varphi$$ $$u_\varphi=c$$ $$u_{\varphi\varphi}=0$$
Is that correct?