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

In this article we consider the Matukuma type equationequation(0.1)Δu+K(r)up=0in RN for positive radially symmetric solutions. We assume that N>2N>2, p>1p>1 and K(r)⩾0K(r)⩾0, for all r⩾0r⩾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.