6 Single-point blow-up for p > n+1

We now steadily deviate from already known results from Chapter 1, since, from now on, p = n+ 1, and the similarity ODE is not variational anymore. Consider ODE (40) in the case p > n + 1, which, in view of the spatial

rescaled variable y in (39), corresponds to a single-point blow-up. Thus, it is crucial that (40), for p = n + 1, is not variational. Therefore, the solutions of (40) can be traced out by complicated shooting and matching procedures, which are still not completely justified. For practical reasons, we will use a continuation in a parameter approach, which allows us to predict solutions by using those in the variational case p = n+ 1. Recall that, for such ODEs, using a standard inverse function theorem is not straightforward at all, since the differential operator in (40) is degenerate and singular. Nevertheless, we will suggest using Schauder’s fixed point theorem and will arrive at some convincing conclusions concerning the solvability and the multiplicity of solutions (the so-called p-branches of solutions).