Other comments.

1. If we work in polar coordinates, fix the angle variable, and look along the radial direction, we find that near the origin w(x,y) decreases because of the low (second) order negative terms, and far from the origin w(x,y) increases because of the higer order positive terms from W(x,y), and in between there is unique minimum. So the situation is very similar to the case of 1 variables for fixed angles.

   It is known that this kind of consideration can be extended and made rigorous. Actually, we make use of aymptotics of the corresponding dynamical system rather than polar coordinates: K.Hattori, T.Hattori, S.Kusuoka, Probability theory and related fields 84 (1990) 1-26.

   The dynamical system point of view gives the existence in D of critical points of w (fixed points of grad W) as a consequence of the standard Fixed Point Theorems. (D is not closed, but one can prove the existence of the invariant compact subset of grad W in D.) It is the uniqueness in D that matters.

2. Prof. Asano at Utsunomiya Univ. told me that the theory of catastrophe may be related. I looked into an introductory part of text book (H. Noguchi, Catastrophe, Science Library 13, Science-sha, 1977 (in Japanese)) to find that, they seem to deal with extrema. It is easy to see that our critical point which is the nearest one to the x-axis, is generically a saddle point rather than extremum. I couldn't follow if the methods develeped in that field can be applicable to saddle points, much less if they can be applicable to the present problem.
3. I once heard that Morse theory deals with saddle points. However, these methods has no relevance to positivity of the coefficients, while our uniqueness conjecture heavily relies on positivity of the coefficients.