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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 $\Omega $ be a bounded domain in $\mathbb{R}^n$, $n \ge 1$, with $C^2$ boundary $\partial \Omega $, and consider the semilinear elliptic boundary value problem

\begin{align*}L u &= \lambda a u + g(\cdot,u)u, \quad \text{in}\ \Omega ,\\ u &= 0, \quad \text{on}\ \partial \Omega , \end{align*}

where $L$ is a uniformly elliptic operator on $\overline{\Omega }$, $a \in C^0(\overline{\Omega })$, $a$ is strictly positive in $\overline{\Omega }$, and the function $g:\overline{\Omega }\times \mathbb{R}\rightarrow \mathbb{R}$ is continuously differentiable, with $g(x,0) = 0$, $x \in \overline{\Omega }$. A well known result of Rabinowitz shows that an unbounded continuum of positive solutions of this problem bifurcates from the principal eigenvalue $\lambda _1$ of the linear problem. We show that under certain oscillation conditions on the nonlinearity $g$, this continuum oscillates about $\lambda _1$, in a certain sense, as it approaches infinity. Hence, in particular, the equation has infinitely many positive solutions for each $\lambda $ in an open interval containing $\lambda _1$.


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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