On the complex structure of positive solutions to Matukuma-type equations

Abstract

In this article we consider the Matukuma type equation

for positive radially symmetric solutions. We assume that $N>2$, $p>1$ and $K(r)⩾0$, for all $r⩾0$. When $K$ satisfies some appropriate monotonicity assumption, the set of positive solutions of (0.1) is well understood. In this work we propose a constructive approach to start the analysis of the structure of the set of positive solutions when this monotonicity assumption fails. We construct some functions $K$ so that the equation exhibits a very complex structure. This function $K$ depends on a set of four parameters: $p$, $N$ and the limits at zero and infinity of certain quotient describing the growth of $K$.