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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