### B P Rynne and M A Youngson

#### Abstract

Let $\Om$ be a bounded domain in $\R^m$, $m>1$, with smooth boundary $\pa \Om$. We consider the boundary value problem \begin{myalign} \L u := - \ds\sum_{i,j=1}^m \dfrac{\pa}{\pa x_i} \left( a_{ij} \dfrac{\pa u}{\pa x_j} \right) + b u = &\la a u + h(\cdot,u,\grad u,\la), \quad {\rm in}\ \Om,\\[1 ex] u = &0, \quad {\rm on}\ \pa\Om, \end{myalign} where $\L$ is uniformly elliptic in $\bOm$ and $a$ is strictly positive. The non-linearity $h$ is continuous and satisfies $$|h(x,\xi,\eta,\la)| \le M_0 |\xi| + M_1 |\eta|, \quad (x,\xi,\eta,\la) \in \bOm \X \R \X \R^m \X \R,$$ as either $|(\xi,\eta)| \tends 0$ or $|(\xi,\eta)| \tends \infty$, for some constants $M_0$, $M_1$ ($|\cdot|$ denotes the Euclidean norm). We show that there exist global sets of non-trivial, positive or negative solutions $(\la,u)$ bifurcating from either $u=0$ or `$u=\infty$' respectively. These sets have similar global properties to those of the bifurcating continua found in Rabinowitz' well known global bifurcation theorem.