[UPDATE] This is the last post for this problem.
A couple of weeks ago, I came up with the solution by using moving frame which was learned in Differential Equation class long time ago. Briefly speaking, it is a good piece of work, in constructing special solution, and a nice try with coordinate transformation.
Now I simply list my trials in probing the path to the solution.
- As I mentioned in last post, I would like to continue using the same technique, with a CGO-like approach, however, after a long time of trials, I gave up with this method. This method will give out a very impressive and concise representation of the problem:
after we multiply to the equation, it will morph into divergence form.
To my knowledge of this degenerate elliptic equation, we need to apply weighted Sobolev space theory, but unfortunately the theory cannot rule out the existence of singularities, and I came up with a counterexample for that. However, I still believe this method can be promising in 2D case.
- For series form, if we require the coefficients of expansion as analytic series, i.e.
This can give a recursion formula for the problem, but the analytic property will force the problem to be unsolvable. What a pity.