Structure theorems for positive radial solutions of the generalized scalar curvature equation, when the curvature makes a finite number of oscillations
by
Matteo Franca
University of Florence, Department of Mathematics

We give some structure results for the positive radial solutions of the following equation:

\Deltapu+K(r) u|u|q-1=0

where K(r) is a function strictly positive and bounded, which makes a finite number of oscillations. Here r=|x|, x in Rⁿ, 2 n/n+2 <= p <= 2, n > p > 1, and q = p^*-1 = (n p)/(n-p)-1. In particular we manage to classify positive regular and singular solutions when K(r) is monotone increasing as r tends to 0, and monotone decreasing as r tends to infinity and it makes a finite number of oscillations. The results are new even when p=2, that is when we consider the usual Laplacian.

The proofs make use of a new Emden-Fowler transform which allow us to consider a 2-dimensional dynamical system thus giving a geometrical point of view on the problem. A key role in the analysis is played by an energy function which is a dynamical interpretation of the Pohozaev function used by Kawano, Ni, Yanagida and Yotsutani, among the others.

Date received: June 1, 2003

Copyright © 2003 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # calo-55.