I am interested in using degree theory to examine some semilinear problems. 
But instead of just looking for solutions lets assume i am looking for a certain class of solutions; for instance lets consider just stable solutions. Is there some way to adjust the usual degree theory so that it can be applied in the restricted class of solutions? I realize this is a somewhat vague question, but any comments would be appreciated. 



degree theory for elliptic equations; special solutions
The whole point of degree theory is its topological invariance. Now consider a saddle node bifurcation where you have a change from no solution to a stable and an unstable solution. This example should be enough to convince you that a "degree" which counts only stable solutions cannot exist.


