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.
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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.
answered Oct 16, 2015 at 2:34
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$\begingroup$ thanks for the comment... (I symbol pushed a bit with some elliptic pde's i know and came to the same conclusion... but i thought there may be some hope for something else.... I guess not). thats for your answer $\endgroup$– Math604Commented Oct 16, 2015 at 3:50