The singular set of
the differential equation
Considering the above equation as a map
Φ : E → F, Φ(u) = = -Δu + u3
– λu, 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
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.