| Abstract: |
| In this talk we present some recent structure results concerning radial
solutions for production/absorption/diffusion
equations, i.e.:
$$ \Delta u + k(|x|) u^{q-1}=0 $$
where $x \in \mathbb{R}^n$, $n>2$, $q>2$ and $k(|x|)$ changes sign.
Namely either we assume $k(r)>0$ (production)
in a ball, and $k(r) < 0$ (absorption) outside and $2 $2^*$.
Our aim is to describe the new scenarios created by
the coexistence of absorption and reaction effects: in particular we
found ground state with fast decay which do not exist if
$k$ is a constant.
The proofs are developed by applying Fowler transformation and
employing dynamical systems techniques.
The results are contained in \textbf{FS1}, \textbf{FS2}: in the former
we consider the simpler case where $k$ is piecewise constant, in
the latter the case where $k$ is smooth.
In \textbf{FS2} we need to perform an extension of the standard
invariant manifold theory for non-autonomous system to the case where
non-continuable trajectories are present: we believe this technique is
of intrinsic mathematical interest.
The argument are generalized to include a spatial dependent Hardy term
too.
|
|