Electron. J. Diff. Equ., Vol. 2010(2010), No. 58, pp. 1-12.

### A global curve of stable, positive solutions for a p-Laplacian problem Bryan P. Rynne

Abstract:
We consider the boundary-value problem

where ( ), , , , and the function is and satisfies

These assumptions on imply that the trivial solution is the only solution with or , and if then any solution is {\em positive}, that is, on .

We prove that the set of nontrivial solutions consists of a curve of positive solutions in , with a parametrisation of the form , where is a function defined on , and is a suitable weighted eigenvalue of the -Laplacian ( may be finite or ), and satisfies

We also show that for each the solution is globally asymptotically stable, with respect to positive solutions (in a suitable sense).

Submitted August 13, 2009. Published April 28, 2010.
Math Subject Classifications: 34B15.
Key Words: Ordinary differential equations; p-Laplacian; nonlinear boundary value problems; positive solutions; stability.

Show me the PDF file (286 KB), TEX file, and other files for this article.

 Bryan P. Rynne Department of Mathematics and the Maxwell Institute for Mathematical Sciences Heriot-Watt University Edinburgh EH14 4AS, Scotland email: bryan@ma.hw.ac.uk