The singular set of the differential equation

Considering the above equation as a map Φ : E → F, Φ(u) = = -Δu + u3λu, where E and F are suitable Banach spaces, the singular set  S  of  Φ consists of the points u in E in which Φ is not invertible, i.e. for which Φ’(u)[v] = -Δv + u2 v – λv = 0 has a nontrivial solution v in E. It is shown in [19]  that for 0 < λ < λ1/7 , where λ1 is the first non-zero eigenvalue of the Laplacian with Neumann boundary conditions, the singular set is diffeomorphic to the unit sphere in E. In addition, all the singular points can be classified: the points on an “equator set” C, i.e. on a set of codimension 1 in the set S (resp. of codimension 2 with respect to E), consist of cusp points, while all the other points S\C consist of fold points.

To understand the solution structure of the above equation, we are interested in Φ(S), the image of the singular set  S under the map Φ. It is proved in [19] that if 0 < λ < λ1/7, then Φ|S is a diffeomorphism. The pictures on the web-page represent (the 3-dimensional projection) of Φ(S). The implication for the partial differential equation is as follows:
- if the forcing term  f  lies inside Φ(S), then the equation has exactly 3 solutions
- if   f  lies outside of  Φ(S), i.e. in F\Φ(S) , then the equation has exactly 1 solution
- if  f  lies on  Φ(S\C), then the equation has exactly 2 solutions

Another way to interpret this result is the following: if we move the parameter λ across the first eigenvalue λ0 = 0, then a bifurcation is taking place. If  λ < λ0 , then the equation has for every  f  a unique solutions, while for  λ > λ0 (and near λ0) we get for some forcing terms (those lying in Φ(S)) three solutions.    

For further reading, see the survey article [37].