Electronic Journal of Differential Equations, Vol. 2000(2000), No. 62, pp. 1–6. ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu ejde.math.unt.edu (login: ftp) A GEOMETRIC ANALYSIS OF A SINGULAR ODE RELATED TO THE STUDY OF A QUASILINEAR PDE JUKKA TUOMELA Abstract. In this note we complement the analysis of a singular ODE given in [2]. Using geometric arguments we are able to settle the structure of the existence and non-existence regions in the parameter space and improve the nonexistence bound. 1. Introduction In [2] existence and uniqueness questions of radial solutions of a class of quasilin- ear elliptic PDEs in a ball, having strong dependence in the gradient, are analysed in terms of the related singular ODE, see also [3]. The authors transform the ODE to an integral equation and look for the fixed points of this operator equation. We complement their analysis using only elementary geometric methods. We show that in addition to the regular solution analysed in [2] there exist also infinitely many singular solutions. It is not quite clear if these ‘new’ solutions can be used to con- struct new solutions to the original PDE, because at least they are too singular to yield strong solutions. All solutions blow up in finite time, and we give a nonex- istence result which for all parameter values is sharper than the one given in [2]. Finally we prove that the regions of existence and nonexistence in the parameter space have a very simple structure, answering rather completely the question raised in [2]. 2. Results In [2] the following ODE is considered t ε ω ′ − g 0 γt γ+ε−1 − f 0 ω δ =0 (1) where f 0 and g 0 are assumed to be positive constants and other parameters will be specified below. We are only interested in nonnegative solutions which satisfy this equation for t> 0 and which can be continuously extended to t = 0. To simplify the study we first observe Lemma 1. Let u be a solution of t ε u ′ − γt γ+ε−1 − u δ =0 (2) Mathematics Subject Classification. 35J60, 35B65, 34C10. Key words. singular ODE, quasilinear PDE. c 2000 Southwest Texas State University. Submitted April 14, 2000. Revised September 12, 2000. Published October 11, 2000. 1