The following theorem is well-known
Theorem (Liouville). Let be a simply connected domain in . Then all real solutions of
in where a constant, are of the form
In geometry, our PDE
says that under the case , it holds
where denotes the standard metric on with constant curvature . Thus we have
Corollary. All solutions of the PDE in with and
are of the form
The proof of this corollary is quite simple.
Proof. By the integrability assumption cannot have an essential singularity at infinity, for otherwise, would conver (possible except one point) infinitely many times near infinity, which is impossible.
for some . By composing with an inversion, we may assume the former case holds. Then maps onto . Since cannot cover (notice that for all ), does not have poles in . This means is a covering map and therefore it assumes the form
for some and . A substitution into our PDE gives the desired conclusion.
Besides, one can see that the intergrability condition is also necessary for asymptotic radial symmetry. All non-radial solutions, which arise from transcendental functions, satisfy
Let’s go back to the theorem, it does not hold for domains which are not simply connected. For instance, the function
is a solution of our PDE in the punctured disc with an isolated singularity at the origin. Such a result for this domain was derived by K.Chou and Y. Wan [here].
A natural extension, but has its own interest, is the so-called Toda system SU(n). The simplest case, known as SU(2), is the following
The solutions of SU(2) are already known, precisely, they can be written as follows
where are real and are complex numbers. We refer the reader to a paper due to del Pino, M. Kowalczyk and J. Wei for details [here].
The following pictures demonstrate the shape of these solutions for a particular choice of eight parameters (i.e. , , ).