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International Journal of Mathematics and Mathematical Sciences
Volume 2011, Article ID 608576, 22 pages
http://dx.doi.org/10.1155/2011/608576
Research Article

Radially Symmetric Solutions of a Nonlinear Elliptic Equation

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

Received 30 December 2010; Accepted 18 April 2011

Academic Editor: Frank Werner

Copyright © 2011 Edward P. Krisner and William C. Troy. 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.

Abstract

We investigate the existence and asymptotic behavior of positive, radially symmetric singular solutions of 𝑀 ξ…ž ξ…ž + ( ( 𝑁 βˆ’ 1 ) / π‘Ÿ ) 𝑀 ξ…ž βˆ’ | 𝑀 | 𝑝 βˆ’ 1 𝑀 = 0 , π‘Ÿ > 0 . We focus on the parameter regime 𝑁 > 2 and 1 < 𝑝 < 𝑁 / ( 𝑁 βˆ’ 2 ) where the equation has the closed form, positive singular solution 𝑀 1 = ( 4 βˆ’ 2 ( 𝑁 βˆ’ 2 ) ( 𝑝 βˆ’ 1 ) / ( 𝑝 βˆ’ 1 ) 2 ) 1 / ( 𝑝 βˆ’ 1 ) π‘Ÿ βˆ’ 2 / ( 𝑝 βˆ’ 1 ) , π‘Ÿ > 0 . Our advance is to develop a technique to efficiently classify the behavior of solutions which are positive on a maximal positive interval ( π‘Ÿ m i n , π‘Ÿ m a x ) . Our approach is to transform the nonautonomous 𝑀 equation into an autonomous ODE. This reduces the problem to analyzing the behavior of solutions in the phase plane of the autonomous equation. We then show how specific solutions of the autonomous equation give rise to the existence of several new families of singular solutions of the 𝑀 equation. Specifically, we prove the existence of a family of singular solutions which exist on the entire interval ( 0 , ∞ ) , and which satisfy 0 < 𝑀 ( π‘Ÿ ) < 𝑀 1 ( π‘Ÿ ) for all π‘Ÿ > 0 . An important open problem for the nonautonomous equation is presented. Its solution would lead to the existence of a new family of “super singular” solutions which lie entirely above 𝑀 1 ( π‘Ÿ ) .