Heriot-Watt Mathematics Report Series
HWM98-35, 22 Oct 1998
Bifurcation of positive solutions from zero or infinity in elliptic problems which are not linearizable
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.
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