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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Oscillating global continua of positive solutions of semilinear elliptic problems
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by Bryan P. Rynne PDF
Proc. Amer. Math. Soc. 128 (2000), 229-236 Request permission

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 $\bar {\Omega }$, $a \in C^0(\bar {\Omega })$, $a$ is strictly positive in $\bar {\Omega }$, and the function $g:\bar {\Omega } \times \mathbb {R} \to \mathbb {R}$ is continuously differentiable, with $g(x,0) = 0$, $x \in \bar {\Omega }$. 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
  • Received by editor(s): March 26, 1998
  • Published electronically: May 27, 1999
  • Communicated by: Lesley M. Sibner
  • © Copyright 1999 American Mathematical Society
  • 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
  • MathSciNet review: 1641097