The singular set of the differential equation
Considering the above equtation
as a map Φ : E → F, 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 the “equator
set” C, i.e. on a set of codimension
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 images
represent (the 3-dimensional projection) of Φ(S). The meaning for the
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 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 three solutions.