## The singular set of
the differential equation

Considering the above equation as a map
Φ : E → F, Φ(u) = = -Δu + u^{3}
– 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 + u^{2} v
– λv = 0 has a nontrivial solution v in E.
It is shown in [19] that for 0 <
λ < λ_{1}/7 , where λ_{1 }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 < λ_{1}/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 λ_{0
}= 0, then a __bifurcation__ is taking place. If λ < λ_{0} , then
the equation has for every *f * a unique solutions, while for λ > λ_{0 }(and near
λ_{0}) we get for some forcing terms (those lying in Φ(S))
three solutions. _{ }

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

##