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International Journal of Differential Equations
Volume 2012, Article ID 296591, 34 pages
Research Article

Radially Symmetric Solutions of

1Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA
2Department of Mathematics, University of Pittsburgh at Greensburg, Greensburg, PA 15601, USA

Received 31 May 2012; Accepted 10 August 2012

Academic Editor: Julio Rossi

Copyright © 2012 William C. Troy and Edward P. Krisner. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We investigate solutions of and focus on the regime and . Our advance is to develop a technique to efficiently classify the behavior of solutions on , their maximal positive interval of existence. Our approach is to transform the nonautonomous equation into an autonomous ODE. This reduces the problem to analyzing the phase plane of the autonomous equation. We prove the existence of new families of solutions of the equation and describe their asymptotic behavior. In the subcritical case there is a well-known closed-form singular solution, , such that as and as . Our advance is to prove the existence of a family of solutions of the subcritical case which satisfies for infinitely many values . At the critical value there is a continuum of positive singular solutions, and a continuum of sign changing singular solutions. In the supercritical regime we prove the existence of a family of “super singular” sign changing singular solutions.