International Journal of Mathematics and Mathematical Sciences
Volume 2011 (2011), Article ID 564930, 14 pages
doi:10.1155/2011/564930
Research Article

Bifurcation of Gradient Mappings Possessing the Palais-Smale Condition

Energy Edge Pty Ltd., P.O. Box 10755, Brisbane, QLD 4000, Australia

Received 30 December 2010; Revised 28 February 2011; Accepted 4 March 2011

Academic Editor: Marco Squassina

Copyright © 2011 Elliot Tonkes. 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

This paper considers bifurcation at the principal eigenvalue of a class of gradient operators which possess the Palais-Smale condition. The existence of the bifurcation branch and the asymptotic nature of the bifurcation is verified by using the compactness in the Palais Smale condition and the order of the nonlinearity in the operator. The main result is applied to estimate the asyptotic behaviour of solutions to a class of semilinear elliptic equations with a critical Sobolev exponent.