The two specific examples of W that I gave generate the renormalization map (RG) (x,y) -> (Wx,Wy) of the self-avoiding walks on the 3 and 4 dimensional Sierpinski gasket. (3dSG: K.Hattori, T.Hattori, S.Kusuoka, Probability theory and related fields 84 (1990) 1-26, 4dSG: T.Hattori, T.Tsuda, in preparation.)
The only way I know of proving the uniqueness of critical point of w in D for these examples is by brute force. Namely, we can prove the uniqueness by eliminating y or x from the fixed point equations, and reduce the problem to finding extremum of a polynomial with one variable. (Note. If you think that this is easy, jsut try! The reduced polynomial generally is a polynomial of very large degree, and the coefficients are complicated functions of the coefficients in the original polynomial W. Hence it is difficult to analyze the reduced polynomial for each specific case, and also difficult to give a proof with indefinite coefficients. On the other hand, in these days of Mathematica and Pentium, one can try as many examples as one likes successfully. But compilation of examples is far from a proof...)
For the 5 and higher dimensional Sierpinski gasket, the corresponding polynomials have 3 or more variables, hence the Problem 1 has no direct applications to the original problems. The main motivation of proposing the Problem 1 is, so to say, phylosophical. It is related to the renormalization group (map) picture for critical phenomena. The dynamical system defined by grad W is, for me, a prototype of RG, with positivity of coefficients implying positivity of statistical weights (or positivity of measures), and the invariant domain D representing a universality class.
All the unsolved problems related to the universality (a notion which, so far, is not much more than at philosophy level), such as universality of critical exponents in statistical mechanics and RG view of unsolved problems in quantum field theories such as the triviality of phi4 theories, triviality or existence of QED, and permanent confinement of non-abelian gauge theories, should, from RG point of view, be solved as a consequence of (global) uniqueness of fixed points in an invariant domain and convergence of RG trajectory to the fixed point.
If there is (and I believe that there is) a general mathematical theory of RG (not just hard analyses for specific models in one extreme, nor hand waving "physics" arguments in other extereme), we must have some general "elegant" solution to the Prolem 1, which is an 2 dimensional analog of RG. The real problem (in statistical mechanics and quantum field theories) of RG are for systems in infinite dimensions. We will have little hope there if we have no hope in just 2 dimensions!