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

  • Moxun Tang

    Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA
  • Patricio Felmer

    Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático (CNRS UMI2807), Universidad de Chile, Casilla 170, Correo 3, Santiago, Chile
  • Alexander Quaas

    Departamento de Matemática, Universidad Técnica Santa María, Casilla V-110, Avda. España 1680, Valparaíso, Chile


In this article we consider the Matukuma type equation

Δu+K(r)up=0in RN\mathrm{\Delta }u + K(r)u^{p} = 0\:\text{in }\mathbb{R}^{N}

for positive radially symmetric solutions. We assume that N>2N > 2, p>1p > 1 and K(r)0K(r)⩾0, for all r0r⩾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.

Cite this article

Moxun Tang, Patricio Felmer, Alexander Quaas, On the complex structure of positive solutions to Matukuma-type equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), no. 3, pp. 869–887

DOI 10.1016/J.ANIHPC.2008.03.006