Heriot-Watt Mathematics Report Series
HWM98-61, 20 Jan 1999
Curves of solutions through points of neutral stability
K J Brown
Abstract
The structure of the set of positive solutions $(\lambda , u)$
of the semilinear elliptic boundary value problem
$$
- \Delta u (x) = \lambda f(u(x)) \quad {\rm for } \quad x \in D; \qquad u(x) = 0
\quad {\rm on} \quad \partial D
$$
where $D$ is a bounded region is investigated. Sufficient conditions are
given to ensure that if $(\lambda , u)$ is such that $u$ is a
positive neutrally stable solution then all solutions in a neighbourhood
of $(\lambda , u)$ lie on a single curve in the $(\lambda , u)$ plane.
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