Heriot-Watt Mathematics Report Series
HWM98-31, 22 Oct 1998
Oscillating global continua of positive solutions of nonlinear elliptic problems
B P Rynne
Abstract
Let $\Om$ be a bounded domain in $\R^n$, $n \ge 1$, with $C^2$ boundary
$\pa \Om$, and consider the semilinear elliptic boundary value problem
\begin{align*}
L u
&= \la a u + g(\cdot,u)u, \quad {\rm in}\ \Om,\\
u &= 0, \quad {\rm on}\ \pa\Om,
\end{align*}
where
$L$ is a uniformly elliptic operator on $\bOm$,
$a \in C^0(\bOm)$, $a$ is strictly positive in $\bOm$,
and the function
$g:\bOm \X \R \tends \R$ is continuously differentiable,
with $g(x,0) = 0$, $x \in \bOm$.
A well known result of Rabinowitz shows that an unbounded continuum of
positive solutions of this problem bifurcates from the principal
eigenvalue $\la_1$ of the linear problem.
We show that under certain oscillation conditions on the nonlinearity $g$,
this continuum oscillates about $\la_1$, in a certain sense, as it approaches infinity.
Hence, in particular, the equation has infinitely many positive solutions
for each $\la$ in an open interval containing $\la_1$.
Google Scholar Search: links, citations and journal (if available)
Contact Details | 1998 Reports Index |
Full Index