Oscillating global continua of positive solutions of semilinear elliptic problems

Author:
Bryan P. Rynne

Journal:
Proc. Amer. Math. Soc. **128** (2000), 229-236

MSC (1991):
Primary 35B32; Secondary 35B65

DOI:
https://doi.org/10.1090/S0002-9939-99-05168-0

Published electronically:
May 27, 1999

MathSciNet review:
1641097

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Abstract: Let be a bounded domain in , , with boundary , and consider the semilinear elliptic boundary value problem

where is a uniformly elliptic operator on , , is strictly positive in , and the function is continuously differentiable, with , . A well known result of Rabinowitz shows that an unbounded continuum of positive solutions of this problem bifurcates from the principal eigenvalue of the linear problem. We show that under certain oscillation conditions on the nonlinearity , this continuum oscillates about , in a certain sense, as it approaches infinity. Hence, in particular, the equation has infinitely many positive solutions for each in an open interval containing .

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Additional Information

**Bryan P. Rynne**

Affiliation:
Department of Mathematics, Heriot-Watt University, Riccarton, Edinburgh EH14 4AS, Scotland

Email:
bryan@ma.hw.ac.uk

DOI:
https://doi.org/10.1090/S0002-9939-99-05168-0

Keywords:
Global bifurcation,
semilinear elliptic equations

Received by editor(s):
March 26, 1998

Published electronically:
May 27, 1999

Communicated by:
Lesley M. Sibner

Article copyright:
© Copyright 1999
American Mathematical Society